title: Alliance 90/The Greens

text: In March 2011 (two weeks after the Fukushima nuclear disaster had begun), the Greens made large gains in Rhineland-Palatinate and in Baden-Württemberg. In Baden-Württemberg they became the senior partner in a governing coalition for the first time. Winfried Kretschmann is now the first Green to serve as Minister-President of a German State (Cabinet Kretschmann I and II). Polling data from August 2011 indicated that one in five Germans supported the Greens. From 4 October 2011 to 4 September 2016, the party was represented in all state parliaments.


title: Alliance 90/The Greens

text: Like the Social Democrats, the Greens backed Chancellor Angela Merkel on most bailout votes in the German parliament during her second term, saying their pro-European stances overrode party politics. Shortly before the elections, the party plummeted to a four-year low in the polls, undermining efforts by Peer Steinbrück's Social Democrats to unseat Merkel.


title: Alliance 90/The Greens

text: While being in opposition on the federal level since 2005, the Greens have established themselves as a powerful force in Germany's political system. By 2016, the Greens had joined 11 out of 16 state governments in a variety of coalitions. Over the years, they have built up an informal structure called G-coordination to organize interests between the federal party office, the parliamentary group in the Bundestag, and the Greens governing on the state level.


title: Alliance 90/The Greens

text: The Greens remained the smallest of six parties in the Bundestag in the 2017 federal election, winning 8.9% of votes. After the election, they entered into talks for a Jamaica coalition with the CDU and FDP. Discussions collapsed after the FDP withdrew in November.


title: Alliance 90/The Greens

text: After the federal election and unsuccessful Jamaica negotiations, the party held elections for two new co-leaders; incumbents Özdemir and Peter did not stand for re-election. Robert Habeck and Annalena Baerbock were elected with 81% and 64% of votes, respectively. Habeck had served as deputy premier and environment minister in Schleswig-Holstein since 2012, while Baerbock had been a leading figure in the party's Brandenburg branch since 2009. Their election was considered a break with tradition, as they were both members of the moderate wing.


title: Alliance 90/The Greens

text: The Greens saw a major surge in support during the Bavarian and Hessian state elections in October 2018, becoming the second largest party in both. They subsequently rose to second place behind the CDU/CSU in national polling, averaging between 17% and 20% over the next six months.


title: Alliance 90/The Greens

text: In the 2019 European Parliament election, the Greens achieved their best ever result in a national election, placing second with 20.5% of the vote and winning 21 seats. National polling released after the election showed a major boost for the party. The first poll after the election, conducted by Forsa, showed the Greens in first place on 27%. This was the first time the Greens had ever been in first place in a national opinion poll, and the first time in the history of the Federal Republic that any party other than the CDU/CSU or SPD had placed first in a national poll. This trend continued as polls from May to July showed the CDU/CSU and Greens trading first place, after which point the CDU/CSU pulled ahead once more. The Greens continued to poll in the low 20% range into early 2020.


title: Alliance 90/The Greens

text: The Greens recorded best-ever results in the Brandenburg (10.8%) and Saxony (8.6%) state elections in September 2019, and subsequently joined coalition governments in both states. They suffered an unexpected decline in the Thuringian election in October, only narrowing retaining their seats with 5.2%. In the February 2020 Hamburg state election, the Greens became the second largest party, winning 24.2% of votes cast.


title: Alliance 90/The Greens

text: In March 2021, the Greens improved their performance in Baden-Württemberg, where they remained the strongest party with 32.6% of votes, and Rhineland-Palatinate, where they moved into third place with 9.3%.


title: Alliance 90/The Greens

text: Due to their sustained position as the second most popular party in national polling ahead of the September 2021 federal election, the Greens chose to forgo the traditional dual lead-candidacy in favour of selecting a single Chancellor candidate. Co-leader Annalena Baerbock was announced as Chancellor candidate on 19 April and formally confirmed on 12 June with 98.5% approval.


title: Alliance 90/The Greens

text: The Greens surged in opinion polls in late April and May, briefly surpassing the CDU as the most popular party in the country, but their numbers slipped back after Baerbock was caught up in several controversies. Her personal popularity also fell below that of both Armin Laschet and Olaf Scholz, the Chancellor candidates for the CDU and SPD, respectively. The party's fortunes did not reverse even after the July floods, which saw climate change return as the most important issue among voters. The situation worsened in August as the SPD surged into first place to the detriment of both the CDU and Greens.


title: Alliance 90/The Greens

text: The Greens finished in third place in the 2021 federal election with 14.8% of votes. Though their best ever federal election result, it was considered a bitter disappointment in light of their polling numbers during the previous three years. They entered coalition talks with the FDP and SPD, eventually joining a traffic light coalition under Chancellor Olaf Scholz which took office on 8 December 2021. The Greens have five ministers in the Scholz cabinet, including Robert Habeck as Vice-Chancellor and Annalena Baerbock as foreign minister.


title: Alliance 90/The Greens

text: Since party statute mandates that party leaders may not hold government office, Baerbock and Habeck stepped down after entering cabinet. At a party conference in January 2022, Ricarda Lang and Omid Nouripour were elected to succeed them. At the time of the her election, Lang was 28 years old, speaker for women's issues, and a former leader of the Green Youth. 46-year-old Nouripour was foreign affairs spokesman and a member of the Bundestag since 2006. Of the new leaders, Lang is considered a representative of the party's left-wing, while Nouripour represents the right-wing.


title: Alliance 90/The Greens

text: The West German Greens played a key role in the development of green politics in Europe, with their original program outlining "four principles: ecological, social, grassroots, and non-violent." Initially ideologically heterogenous, the party took up a position on the radical left in its early years, which were dominated by conflicts between the more left-wing "Fundi" (fundamentalist) and more moderate "Realo" (realist) factions. These conflicts became less significant as the party moved toward the political mainstream in the 1990s.


title: Alliance 90/The Greens

text: In the 21st century, Alliance 90/The Greens is typically classified on the centre-left of the political spectrum, focusing on environmentalist and socially progressive policies. Emphasis is placed on mitigating climate change, reducing carbon emissions, and fostering sustainability and environmentally-friendly practices. They support equality, social justice, and humanitarian responses to events such as the European migrant crisis. Their fiscal platform is flexible and seeks to balance social, economic, and environmental interests. The party is strongly pro-European, advocating European federalism, and promotes wider international cooperation, including strengthening existing alliances.


title: Alliance 90/The Greens

text: Starting from the leadership of Annalena Baerbock and Robert Habeck, commentators have observed the Greens taking a pragmatic, moderate approach to work with parties from across the political spectrum. Baerbock described their stances and style as a form of "radical realism" attempting to reconcile principles with practical politics. At the same time, the party has denounced populism and division, and placed rhetorical emphasis on optimism and cross-party cooperation. Accompanied by record high popularity and election results, this led some to suggest that the Greens were filling a gap in the political centre left by the declining popularity of the CDU/CSU and SPD.


title: Alliance 90/The Greens

text: The party supports the legalization and regulation of cannabis and is the sponsor of the proposed German cannabis control bill.


title: Alliance 90/The Greens

text: The Greens are regarded as taking a centrist line on defense and pushing for a stronger common EU foreign policy, especially against Russia and China. Green Party co-leader Annalena Baerbock has proposed a post-pacifist foreign policy. She supports eastward expansion of NATO and has considered the number of UN resolutions critical of Israel as "absurd compared to resolutions against other states." The party's program included references to NATO as an "indispensable" part of European security. The Greens have promised to abolish the contested Nord Stream 2 pipeline to ship Russian natural gas to Germany. The party criticized the EU's investment deal with China. In 2016, the Greens criticised Germany's defense plan with Saudi Arabia, which has been waging war in Yemen and has been accused of massive human rights violations.


title: Alliance 90/The Greens

text: The party remains divided over issues such as nuclear disarmament and U.S. nuclear weapons on German territory. Some Greens want Germany to sign the United Nations' Treaty on the Prohibition of Nuclear Weapons.


title: Alliance 90/The Greens

text: Ever since the party's inception, The Greens have been concerned with the immediate halt of construction or operation of all nuclear power stations. As an alternative, they promote a shift to non-nuclear renewable energy and a comprehensive program of energy conservation.


title: Alliance 90/The Greens

text: In 1986, large parts of Germany were covered with radioactive contamination from the Chernobyl disaster and Germans went to great lengths to deal with the contamination. Germany's anti-nuclear stance was strengthened. From the mid-1990s onwards, anti-nuclear protests were primarily directed against transports of radioactive waste in "CASTOR" containers.


title: Alliance 90/The Greens

text: After the Chernobyl disaster, the Greens became more radicalised and resisted compromise on the nuclear issue. During the 1990s, a re-orientation towards a moderate program occurred, with concern about global warming and ozone depletion taking a more prominent role. During the federal red-green government (1998–2005) many people became disappointed with what they saw as excessive compromise on key Greens policies.


title: Alliance 90/The Greens

text: Energy policy is still the most important cross-cutting issue in climate and economic policies. Implementation of Green Policy would see electricity generation from 100 percent renewable sources as early as 2040. The development of renewable energy and combined heat and power is also a great opportunity for technical and economic innovation. Solar industry and environmental technologies are already a significant part of key industries providing jobs which need to be developed and promoted vigorously. In addition, a priority of green energy policy is increasing the thermal insulation and energy efficiency of homes, the phaseout of all nuclear energy generation with possible high-efficiency gas-fired power plants operational during the transition phase.


title: Alliance 90/The Greens

text: The central idea of green politics is sustainable development. The concept of environmental protection is the cornerstone of Alliance 90/The Greens policy. In particular, the economic, energy and transport policy claims are in close interaction with environmental considerations. The Greens acknowledge the natural environment as a high priority and animal protection should be enshrined as a national objective in constitutional law. An effective environmental policy would be based on a common environmental code, with the urgent integration of a climate change bill. During the red-green coalition (1998–2005) a policy of agricultural change was launched labeled as a paradigm shift in agricultural policy towards a more ecological friendly agriculture, which needs to continue.


title: Alliance 90/The Greens

text: The Greens have praised the European Green Deal, which aims to make the EU climate neutral by 2050. Climate change is at the center of all policy considerations. This includes environmental policy and safety and social aspects. The plans of the Alliance 90/The Greens provide a climate change bill laying down binding reductions to greenhouse gas emissions in Germany by 2020 restricting emissions to minus 40 percent compared to 1990.


title: Alliance 90/The Greens

text: Alliance 90/The Greens supports the eventual federalization of the European Union into a Federal European Republic (German: "Föderale Europäische Republik"), i.e. a single federal European sovereign state.


title: Alliance 90/The Greens

text: A similarly high priority is given to transport policy. The switch from a traveling allowance to a mobility allowance, which is paid regardless of income to all employees, replacing company car privileges. The truck toll will act as a climate protection instrument internalizing the external costs of transport. Railway should be promoted in order to achieve the desired environmental objectives and the comprehensive care of customers. The railway infrastructure is to remain permanently in the public sector, allowing a reduction in expenditure on road construction infrastructure. The Greens want to control privileges on kerosene and for international flights, introduce an air ticket levy. Restrict speeds nationwide on the highways to 120 km/h and country roads to 80 km/h. The Greens want to create a market incentive and research program of €500 million annually to ensure that by 2020 there are at least two million electric cars on German roads.


title: Alliance 90/The Greens

text: For many years, the Green Party has advocated against the "Ehegattensplitting" policy, under which the incomes of married couples are split for taxation purposes. Furthermore, the Party advocates for a massive increase in federal spending for places in preschools, and for increased investment in education: an additional 1 billion Euros for vocational schools and 200 million Euros more BAföG (Bundesausbildungsförderungsgesetz in German, approximately translated to "the Federal Law for the Advancement of Education") for adults.


title: Alliance 90/The Greens

text: In its 2013 platform, the Green Party successfully advocated for a minimum wage of 8.50 Euro per hour, which was implemented on 1 January 2015. It continues to press for higher minimum wages.


title: Alliance 90/The Greens

text: The Greens want the starting retirement age to remain 67, but with some qualificationsfor example, a provision for partial retirement.


title: Alliance 90/The Greens

text: The Greens advocate decriminalizing marijuana usage and the private growing of plants. Furthermore, the Greens support research on the drug and the use of marijuana for medicinal purposes.


title: Alliance 90/The Greens

text: The Green Party supports the implementation of quotas in executive boards, the policy of equal pay for equal work, and continuing the fight against domestic violence. According to its website, the Green Party "fights for the acceptance and against the exclusion of homosexuals, bisexuals, intersex- and transgender people and others".


title: Alliance 90/The Greens

text: In order to recognize the political persecution that LGBT+ people face abroad, the Green Party wants to extend asylum to LGBTQIA+ people abroad. The policy change was sponsored primarily by Volker Beck, one of the Party's most prominent gay members. Because of the extensive support the Green Party has given the LGBTQIA+ community since its conception, many LGBTQIA+ people vote for the Green Party even if their political ideology does not quite align otherwise.


title: Alliance 90/The Greens

text: A 2000 study by the Infratest Dimap political research company has suggested the Green voter demographic includes those on higher incomes (e.g. above €2000/month) and the party's support is less among households with lower incomes. The same polling research also concluded that the Greens received fewer votes from the unemployed and general working population, with business people favouring the party as well as the centre-right liberal Free Democratic Party. According to Infratest Dimap the Greens received more voters from the age group 34–42 than any other age group and that the young were generally more supportive of the party than the old. (Source: Intrafest Dimap political research company for the ARD.)


title: Alliance 90/The Greens

text: The Greens have a higher voter demographic in urban areas than rural areas, except for a small number of rural areas with pressing local environmental concerns, such as strip mining or radioactive waste deposits. The cities of Bonn, Cologne, Stuttgart, Berlin, Hamburg, Frankfurt and Munich have among the highest percentages of Green voters in the country. The towns of Aachen, Bonn, Darmstadt, Hanover, Mönchengladbach and Wuppertal have Green mayors. The party has a lower level of support in the states of the former German Democratic Republic (East Germany); nonetheless, the party is currently represented in every state landtag except Saarland.


title: Matt LaFleur

text: Matthew Patrick LaFleur ( ; born November 3, 1979) is an American football coach who is the head coach for the Green Bay Packers of the National Football League (NFL). He has served as the quarterback coach of the Washington Redskins, Notre Dame, and the Atlanta Falcons and has been the offensive coordinator for the Los Angeles Rams and the Tennessee Titans.


title: Matt LaFleur

text: Born and raised in Mount Pleasant, Michigan, LaFleur attended Western Michigan University in Kalamazoo and played wide receiver in 1998 and 1999. to Saginaw Valley State in Division II, and he played quarterback from 2000 to 2002, guiding the Cardinals to the D-II playoffs LaFleur left Saginaw Valley State as their all-time leader in passing yards, completions, and passing touchdowns. LaFleur was inducted into the SVSU Cardinal Athletic Hall of Fame on October 1, 2021.


title: Matt LaFleur

text: LaFleur's coaching career began in 2003 at his alma mater, Saginaw Valley State, as an offensive graduate assistant, during which time he also substitute taught in the local high schools, specializing in math. LaFleur proceeded to Central Michigan University in Mount Pleasant and assisted the offensive staff in 2004 and 2005. In 2006, he coached quarterbacks and receivers at Northern Michigan University in Marquette. LaFleur was the offensive coordinator at Ashland University in Ohio in 2007.


title: Matt LaFleur

text: LaFleur then was hired in 2008 by the Houston Texans to serve as an offensive assistant. He assisted both wide receivers and quarterbacks during his two-year stint there and developed a close relationship with offensive coordinator Kyle Shanahan.


title: Matt LaFleur

text: When Shanahan's father Mike was hired by the Washington Redskins, Kyle brought LaFleur to Washington to coach the quarterbacks in 2010. From 2011 to 2013, LaFleur worked alongside 3 other future head coaches in Washington; Kyle Shanahan, Sean McVay, and Mike McDaniel. A primary responsibility of LaFleur's for the 2012 season was to mentor rookie quarterbacks Robert Griffin III and Kirk Cousins.


title: Matt LaFleur

text: After six years of coaching in the NFL, LaFleur returned to college football as the quarterbacks coach at the University of Notre Dame in 2014. LaFleur tutored senior quarterback Everett Golson who posted 3,445 yards and 29 touchdowns through the air, helping him become just the fourth quarterback in school history to eclipse such totals in a single season.


title: Matt LaFleur

text: On February 5, 2015, LaFleur returned to the NFL as the quarterbacks coach for the Atlanta Falcons working under offensive coordinator Kyle Shanahan, with whom he had previously worked with while with the Washington Redskins and Houston Texans. LaFleur's younger brother, Mike, was also an offensive assistant with the Falcons.


title: Matt LaFleur

text: In 2016, LaFleur coached Matt Ryan on his way to winning his sole NFL MVP award. The Falcons reached Super Bowl LI, where they faced the New England Patriots, but squandered a large lead and lost in overtime.


title: Matt LaFleur

text: On February 8, 2017, LaFleur joined the Los Angeles Rams coaching staff as offensive coordinator, working under head coach Sean McVay, with whom he had previously worked during his tenure with the Washington Redskins. Under LaFleur and McVay, the Rams finished the year with an 11–5 record and as the league's number one scoring offense, scoring 478 points through 16 games.


title: Matt LaFleur

text: On January 30, 2018, LaFleur left his position with the Rams to take the same position with the Tennessee Titans. Joining new head coach Mike Vrabel, LaFleur's role in directing the offense increased, as he had play-calling responsibilities unlike during his tenure with the Rams. The season was plagued with injuries, with the Titans losing star tight end Delanie Walker in Week 1, and Mariota dealing with an incessant nerve injury throughout the season. LaFleur and the Titans finished the season with the 27th ranked scoring offense in the NFL.


title: Matt LaFleur

text: LaFleur was hired as the head coach of the Green Bay Packers on On May 30, 2019, LaFleur suffered a torn Achilles while playing basketball. LaFleur also led the Packers to a 2–2 preseason.


title: Matt LaFleur

text: On September 5, 2019, LaFleur made his regular-season head coaching debut against the Chicago Bears, and led the Packers to a 10–3 victory. He also became the first Green Bay coach to win his first game against the Bears since Vince Lombardi in 1959. The Packers finished with a 13–3 record in LaFleur's first season as a head coach, and in the process, LaFleur became the first Packers rookie head coach to win 10 games, make the playoffs, and win the NFC North (as well as going 6–0 in division play in the process). In addition, the 13 wins were the most from a rookie coach since Jim Harbaugh did it with the 49ers in 2011.


title: Matt LaFleur

text: LaFleur led Green Bay to their first postseason berth since the 2016 season as the NFC's number two seed, as he won his postseason debut against the Seattle Seahawks 28–23 in the divisional round of the playoffs to advance to the NFC Championship Game, where the Packers fell to the top-seeded San Francisco 49ers 37–20.


title: Matt LaFleur

text: The Packers began the 2020 season by winning all four games prior to an early Week 5 bye. After a 5–1 start, LaFleur lost his first-ever divisional game, with a 22–28 home loss to the Minnesota Vikings. Despite this, the Packers continued rolling, and LaFleur clinched his second consecutive NFC North title and playoff berth in Week 14 with a 31–24 victory against the Detroit Lions. With a 35–16 win over the Chicago Bears in Week 17, Green Bay clinched the NFC's number one seed and home-field advantage throughout the playoffs for the first time since the 2011 season. The Packers also closed the season out on a six-game winning streak, and LaFleur extended his win–loss record in December games to 9–0.


title: Matt LaFleur

text: LaFleur finished with a 26–6 record in his first two seasons as coach, tied for the second-best start for a coach since the 1970 merger (only George Seifert, at 28–4, has a better record). His team finished as the league's number one offense, scoring 509 points over the course of the season, and with quarterback Aaron Rodgers throwing for a Packers franchise-record 48 touchdowns.


title: Matt LaFleur

text: In the playoffs, LaFleur and the Packers hosted the Los Angeles Rams, who had the league's number one scoring defense and were led by one of LaFleur's former bosses, Rams coach Sean McVay. The Packers defeated the Rams, 32–18, to host the NFC Championship Game for the first time since 2007, where they lost to the eventual Super Bowl champion Tampa Bay Buccaneers, 31–26. LaFleur was criticized for his decision to kick a field goal when the Packers were down 31-23 instead of trying to tie the game with a touchdown and a two point conversion under league MVP Aaron Rodgers.


title: Matt LaFleur

text: LaFleur's 2021 season began with a 38–3 loss to the New Orleans Saints. They proceeded to win each of their next six games to bring the 6–1 Packers to a Thursday night showdown with the NFL's last unbeaten team, the 7–0 Arizona Cardinals. In addition to playing on the road on a short week, the Packers were forced to play without their top three wide receivers, after Davante Adams and Allen Lazard tested positive for COVID-19 and Marquez Valdes-Scantling remained out with a hamstring injury since Week 3. Despite the challenges, the Packers emerged victorious, 24–21 with a thrilling last-minute interception of Cardinals quarterback Kyler Murray, and LaFleur became the winningest head coach through his first 40 career games in NFL history.


title: Matt LaFleur

text: The following week, star quarterback Aaron Rodgers tested positive for COVID-19, and LaFleur named second-year quarterback Jordan Love the starter for their Week 9 game against the Kansas City Chiefs. Although Love was relentlessly blitzed during his first career start, he managed to complete 19 of 34 passes for 190 yards, 1 touchdown and 1 interception. LaFleur took the blame for not being able to counter the Chiefs' aggressive defensive strategies. Special teams miscues ultimately cost the Packers the game, falling 7–13 to drop them to 7–2.


title: Matt LaFleur

text: The Packers then shut out Seattle Seahawks quarterback Russell Wilson the following week, the first such occurrence in his career, to win their Week 10 game 17–0. In Week 11, LaFleur's Packers fell 31–34 to their division rival Minnesota Vikings, but rebounded to win 36–28 against LaFleur's former boss Sean McVay and the Los Angeles Rams the following week. The Packers clinched their third consecutive NFC North title with a tight 31–30 Week 15 victory over the Baltimore Ravens. LaFleur became the first head coach since Dallas's Barry Switzer to win the division in each of his first three years as head coach.


title: Matt LaFleur

text: With a 37–10 Week 17 victory over the Minnesota Vikings, LaFleur's Packers clinched home-field advantage throughout the NFC playoffs and became the first team in NFL history to win 13 games a season in 3 consecutive years. In addition, LaFleur broke a 30-year-old record by George Seifert for the most wins by a head coach (39) through his first 3 seasons in the league. The win also meant to this point in his career, LaFleur had never lost consecutive regular season games during his 3-year tenure.


title: Matt LaFleur

text: The Packers dealt with a notable amount of injuries throughout the season. Top pass-rusher Za'Darius Smith had not played since Week 1, while top cornerback Jaire Alexander had been out since spraining his shoulder in Week 4. Top tackle David Bakhtiari tore his ACL during the 2020 season, and didn't return until Week 18. His replacement, offensive lineman Elgton Jenkins, was lost for the season with a similar ACL injury in Week 11. In addition, starting tight end Robert Tonyan met a similar fate during the Packers' Week 8 showdown in Arizona.


title: Matt LaFleur

text: At the end of the regular season, quarterback Aaron Rodgers was named NFL MVP for the second consecutive year and for the fourth time in his career. Rodgers noted LaFleur's contributions to his winning MVP in 2 of LaFleur's 3 seasons as head coach. Rodgers became the first player in the NFL to win consecutive MVP awards in 12 years, when Peyton Manning won in 2008 and 2009. In the playoffs, they lost their first game to the San Francisco 49ers, 13–10, giving LaFleur a 2–3 postseason record and no Super Bowl appearances in their three straight 13-win seasons. He took the blame for the Packers’ poor special teams effort, particularly one play where they were missing a man on the field during San Francisco's game-winning field goal.


title: Matt LaFleur

text: LaFleur was married to BreAnne Maak, whom he met in college, and they have two sons, Luke and Ty. His younger brother, Mike, is the offensive coordinator for the New York Jets. Robert Saleh was the best man at LaFleur's wedding, as the two became close while working as graduate assistants at Central Michigan.


title: Esther Rolle

text: Esther Elizabeth Rolle (November 8, 1920 – November 17, 1998) was an American actress. She is best known for her role as Florida Evans, on the CBS television sitcom "Maude," for two seasons (1972–1974), and its spin-off series "Good Times", for five seasons (1974–77, 1978–79), for which Rolle was nominated for a Golden Globe Award for Best Actress - Television Series Musical or Comedy in 1976. In 1979, Rolle won the Emmy Award for Outstanding Supporting Actress in a Limited Series or Special for the television film "Summer of My German Soldier".


title: Esther Rolle

text: Esther Rolle was born in Pompano Beach, Florida to Bahamian immigrants Jonathan Rolle (1883–1953), a farmer, and Elizabeth Iris Rolle (née Dames; 1887–1981). Her parents were both born and raised in Nassau, New Providence, The Bahamas and moved to Florida some time after their marriage. She was the tenth of 18 children (children who included siblings and fellow actresses Estelle Evans and Rosanna Carter). Rolle graduated from Blanche Ely High School in Pompano Beach, Florida. She initially studied at Spelman College in Atlanta, before moving to New York City. While in New York, she attended Hunter College before transferring to The New School and then Yale University in New Haven, Connecticut. For many years, Rolle worked in a traditional day job in New York City's garment district.


title: Esther Rolle

text: Rolle was a member of Asadata Dafora's dance troupe, Shogolo Oloba (later renamed the Federal Theater African Dance Troupe). She became the troupe's director in 1960. Rolle's earliest roles were on the stage; her New York stage debut was in the 1962 play "The Blacks". She was often cast in plays produced by Robert Hooks and the Negro Ensemble Company. She also appeared in productions of "The Crucible" and "Blues for Mr. Charlie". Rolle's most prominent early role was as Miss Maybell in the Melvin Van Peebles musical film "Don't Play Us Cheap", and it's subsequent stage adaptation. In 1977, Rolle portrayed Lady Macbeth in Orson Welles' Haitian-influenced version of William Shakespeare's "Macbeth" at the Henry Street New Federal Theater in Manhattan.


title: Esther Rolle

text: Rolle is best known for her television role as Florida Evans, the character she played on two 1970s sitcoms. The character was introduced as Maude Findlay's housekeeper on "Maude", and was spun off in the show's second season into "Good Times", a show about Florida's family. Rolle was nominated in 1975 for the Best Actress in a Musical/Comedy Golden Globe Award for her role in "Good Times". Rolle was 19 years older than the actor (John Amos) who played her husband James (named Henry on Maude) Evans. The James Evans character was only added after Esther Rolle fought hard for a father figure and husband to be added to the show. Rolle had fought for the father character on the show, more relevant themes and scripts and was unhappy with the success of Jimmie Walker's character, J.J. Evans, which she believed took the show in a frivolous direction. John Amos agreed with Rolle about Walker's character and was fired from the show after the third season ended. Later on, in a stand-off with "Good Times" producer Norman Lear, Rolle also quit when her contract ended. Although the show continued without her for the fifth season, she returned for the show's final season. In 1979 she won an Emmy for her role in "Summer of My German Soldier", a made-for-television movie.


title: Esther Rolle

text: Among her guest-starring roles was one on "The Incredible Hulk" in an episode entitled "Behind the Wheel", in which she played a taxicab business owner. In the 1990s, Rolle was a surprise guest on RuPaul's VH-1 talk show. Her "Maude" co-star Bea Arthur was the guest, and Rolle was brought out to surprise Arthur. The two had not seen each other in years, Arthur said, and embraced warmly. Rolle also appeared in a series of psychic hotline TV commercials in the 1990s. "Tell them Esther sent you," was her trademark line.


title: Esther Rolle

text: Rolle released an album of music titled "The Garden of My Mind" in 1975. Rolle's first screen appearance is a small, uncredited role in "To Kill a Mockingbird" (1962), and she later appeared in Gordon Parks' "The Learning Tree" (1969). Her sister, actress Estelle Evans, appeared in both films as well. Esther Rolle appeared early in her career in the film "Nothing But a Man" (1964). After "Good Times" ended, she appeared in a number of made-for-television movies and films, including "Driving Miss Daisy" and "My Fellow Americans". A memorable role was that of Aunt Sarah in the film "Rosewood" (1997). She had a major role in "I Know Why the Caged Bird Sings" based on Maya Angelou's memoir of the same name, and has the distinction of having won the first Emmy Award for the category Outstanding Supporting Actress in a Miniseries or Movie, in 1979, for her work in the television movie "Summer of My German Soldier". She is also credited for her role in the film "The Mighty Quinn" (1989), starring Denzel Washington and Sheryl Lee Ralph, and featuring Robert Townsend. Her last film, "Train Ride", was released in 2000 despite being filmed in 1998.


title: Esther Rolle

text: Rolle's only marriage was to Oscar Robinson. The two were married from 1955 to 1975. They had no children.


title: Esther Rolle

text: Rolle died on November 17, 1998, in Culver City, California, from complications of diabetes, nine days after her 78th birthday. A devout member of the African Methodist Episcopal Church, Rolle requested that her funeral be held at Bethel African Methodist Episcopal Church. She is buried in Westview Community Cemetery in Pompano Beach, Florida.


title: Esther Rolle

text: The cemetery is a historically black burial ground created in 1952, a time when the laws and customs of Florida did not permit white people and black people to be buried in the same cemetery.


title: Esther Rolle

text: Rolle's family donated over 100 items of hers to the African-American Research Library and Cultural Center in Fort Lauderdale, Florida. The collection includes gowns, a black Raggedy Andy doll she endorsed, a recording of poems recited by Rolle, and awards such as the 1974 NAACP Eighth Image Award for Best Actress in a Series and her 1979 Emmy for her role in "Summer of My German Soldier".


title: S. Robson Walton

text: Samuel Robson “Rob” Walton (born October 28, 1944) is an American billionaire heir to the fortune of Walmart, the world's largest retailer. He is the eldest son of Helen Walton and Sam Walton, and was chairman of Walmart from 1992 to 2015. As of October 2022, Walton had an estimated net worth of US$61 billion, making him the 18th richest person in the world. He is also the principal owner of the National Football League (NFL)'s Denver Broncos.


title: S. Robson Walton

text: Rob Walton was born on October 28, 1944, in Tulsa, Oklahoma, the oldest of four children of Sam Walton (1918–1992), cofounder of Walmart, and Helen Walton (1919–2007). He has a younger brother, Jim Walton, and a younger sister, Alice Walton. Another brother, John Walton, died in 2005.


title: S. Robson Walton

text: Walton attended The College of Wooster and graduated from the University of Arkansas in 1966 with a Bachelor of Science degree in business administration, where he was a member of the Lambda Chi Alpha fraternity. He received his juris doctor degree from Columbia Law School in 1969. Walton is also a trustee at The College of Wooster.


title: S. Robson Walton

text: After graduation, Walton became a member of the law firm that represented Walmart, Conner & Winters in Tulsa, Oklahoma. In 1978, he left Tulsa to join Walmart as a senior vice president, and in 1982, he was appointed vice chairman. He was named chairman of the board of directors on April 7, 1992, two days after his father's death.


title: S. Robson Walton

text: Along with his siblings, he has pledged about $2 billion to the Walton Family Foundation from 2008 to 2013.


title: S. Robson Walton

text: On June 7, 2022, an ownership group led by Walton entered into an agreement to purchase the Denver Broncos from the estate of Pat Bowlen for $4.65 billion, which set the record for the most expensive sale of a sports franchise in history, subject to approval from the NFL's finance committee and a 3/4 majority of the full NFL ownership group. Former Secretary of State Condoleezza Rice was announced as added to the ownership group on July 11, 2022. The family then announced on August 2, 2022, that Lewis Hamilton would be added to the ownership group.


title: S. Robson Walton

text: By the time he left Tulsa in 1978, Walton had three children, was divorced from his first wife, and had remarried to Carolyn Funk. He and Carolyn filed for divorce in 2000. He married his third wife Melani Lowman-Walton in 2005. His children include Carrie Walton Penner.


title: S. Robson Walton

text: In 2022, due to a decline in Wal-Mart's share price, Rob and his siblings lost more than $27 billion.


title: S. Robson Walton

text: Walton is a well-known collector of automobiles. His car collection is valued in excess of US$300 million and includes:


title: Ring (mathematics)

text: In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a "ring" is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.


title: Ring (mathematics)

text: Formally, a "ring" is an abelian group whose operation is called "addition", with a second binary operation called "multiplication" that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term "" with a missing i to refer to the more general structure that omits this last requirement; see .)


title: Ring (mathematics)

text: Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has profound implications on its behavior. Commutative algebra, the theory of commutative rings, is a major branch of ring theory. Its development has been greatly influenced by problems and ideas of algebraic number theory and algebraic geometry. The simplest commutative rings are those that admit division by non-zero elements; such rings are called fields.


title: Ring (mathematics)

text: Examples of commutative rings include the set of integers with their standard addition and multiplication, the set of polynomials with their addition and multiplication, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. Examples of noncommutative rings include the ring of real square matrices with , group rings in representation theory, operator algebras in functional analysis, rings of differential operators, and cohomology rings in topology.


title: Ring (mathematics)

text: The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by Dedekind, Hilbert, Fraenkel, and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. They later proved useful in other branches of mathematics such as geometry and analysis.


title: Ring (mathematics)

text: A ring is a set "R" equipped with two binary operations + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms


title: Ring (mathematics)

text: In the terminology of this article, a ring is defined to have a multiplicative identity, while a structure with the same axiomatic definition but without the requirement for a multiplicative identity is instead called a rng (IPA: ). For example, the set of even integers with the usual + and ⋅ is a rng, but not a ring. As explained in "" below, many authors apply the term "ring" without requiring a multiplicative identity.


title: Ring (mathematics)

text: Although ring addition is commutative, ring multiplication is not required to be commutative: "ab" need not necessarily equal "ba". Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called "commutative rings". Books on commutative algebra or algebraic geometry often adopt the convention that "ring" means "commutative ring", to simplify terminology.


title: Ring (mathematics)

text: In a ring, multiplicative inverses are not required to exist. A nonzero commutative ring in which every nonzero element has a multiplicative inverse is called a field.


title: Ring (mathematics)

text: The additive group of a ring is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms. The proof makes use of the "1", and does not work in a rng. (For a rng, omitting the axiom of commutativity of addition leaves it inferable from the remaining rng assumptions only for elements that are products: .)


title: Ring (mathematics)

text: Although most modern authors use the term "ring" as defined here, there are a few who use the term to refer to more general structures in which there is no requirement for multiplication to be associative. For these authors, every algebra is a "ring".


title: Ring (mathematics)

text: The axioms of a ring were elaborated as a generalization of familiar properties of addition and multiplication of integers.


title: Ring (mathematics)

text: Then Z/4Z is a ring: each axiom follows from the corresponding axiom for Z. If "x" is an integer, the remainder of "x" when divided by 4 may be considered as an element of Z/4Z, and this element is often denoted by or formula_9, which is consistent with the notation for 0, 1, 2, 3. The additive inverse of any formula_9 in Z/4Z is formula_11. For example, formula_12


title: Ring (mathematics)

text: With the operations of matrix addition and matrix multiplication, formula_14 satisfies the above ring axioms. The element formula_15 is the multiplicative identity of the ring. If formula_16 and formula_17, then formula_18 while formula_19; this example shows that the ring is noncommutative.


title: Ring (mathematics)

text: More generally, for any ring , commutative or not, and any nonnegative integer , the square matrices of dimension with entries in form a ring: see Matrix ring.


title: Ring (mathematics)

text: The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. In 1871, Richard Dedekind defined the concept of the ring of integers of a number field. In this context, he introduced the terms "ideal" (inspired by Ernst Kummer's notion of ideal number) and "module" and studied their properties. Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting.


title: Ring (mathematics)

text: The term "Zahlring" (number ring) was coined by David Hilbert in 1892 and published in 1897. In 19th century German, the word "Ring" could mean "association", which is still used today in English in a limited sense (for example, spy ring), so if that were the etymology then it would be similar to the way "group" entered mathematics by being a non-technical word for "collection of related things". According to Harvey Cohn, Hilbert used the term for a ring that had the property of "circling directly back" to an element of itself (in the sense of an equivalence). Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers "cycle back". For instance, if then , , , , , and so on; in general, "a" is going to be an integral linear combination of 1, "a", and "a".


title: Ring (mathematics)

text: The first axiomatic definition of a ring was given by Adolf Fraenkel in 1915, but his axioms were stricter than those in the modern definition. For instance, he required every non-zero-divisor to have a multiplicative inverse. In 1921, Emmy Noether gave a modern axiomatic definition of commutative rings (with and without 1) and developed the foundations of commutative ring theory in her paper "Idealtheorie in Ringbereichen".


title: Ring (mathematics)

text: Fraenkel's axioms for a "ring" included that of a multiplicative identity, whereas Noether's did not.


title: Ring (mathematics)

text: Most or all books on algebra up to around 1960 followed Noether's convention of not requiring a 1 for a "ring". Starting in the 1960s, it became increasingly common to see books including the existence of 1 in the definition of "ring", especially in advanced books by notable authors such as Artin, Atiyah and MacDonald, Bourbaki, Eisenbud, and Lang. There are also books published as late as 2006 that use the term without the requirement for a 1.


title: Ring (mathematics)

text: Gardner and Wiegandt assert that, when dealing with several objects in the category of rings (as opposed to working with a fixed ring), if one requires all rings to have a 1, then some consequences include the lack of existence of infinite direct sums of rings, and that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable." Poonen makes the counterargument that the natural notion for rings is the direct product rather than the direct sum. He further argues that rings without a multiplicative identity are not totally associative (the product of any finite sequence of ring elements, including the empty sequence, is well-defined, independent of the order of operations) and writes "the natural extension of associativity demands that rings should contain an empty product, so it is natural to require rings to have a 1".


title: Ring (mathematics)

text: Authors who follow either convention for the use of the term "ring" may use one of the following terms to refer to objects satisfying the other convention:


title: Ring (mathematics)

text: For each nonnegative integer , given a sequence formula_25 of elements of , one can define the product formula_26 recursively: let and let for .


title: Ring (mathematics)

text: As a special case, one can define nonnegative integer powers of an element of a ring: and for . Then for all .


title: Ring (mathematics)

text: A left zero divisor of a ring formula_27 is an element formula_28 in the ring such that there exists a nonzero element formula_29 of formula_27 such that formula_31. A right zero divisor is defined similarly.


title: Ring (mathematics)

text: A nilpotent element is an element formula_28 such that formula_33 for some formula_34. One example of a nilpotent element is a nilpotent matrix. A nilpotent element in a nonzero ring is necessarily a zero divisor.


title: Ring (mathematics)

text: An idempotent formula_35 is an element such that formula_36. One example of an idempotent element is a projection in linear algebra.


title: Ring (mathematics)

text: A unit is an element formula_28 having a multiplicative inverse; in this case the inverse is unique, and is denoted by formula_38. The set of units of a ring is a group under ring multiplication; this group is denoted by formula_39 or formula_40 or formula_41. For example, if "R" is the ring of all square matrices of size "n" over a field, then formula_39 consists of the set of all invertible matrices of size "n", and is called the general linear group.


title: Ring (mathematics)

text: For example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z["X"] (in both cases, Z contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers 2Z does not contain the identity element 1 and thus does not qualify as a subring of Z; one could call 2Z a subrng, however.


title: Ring (mathematics)

text: An intersection of subrings is a subring. Given a subset "E" of "R", the smallest subring of "R" containing "E" is the intersection of all subrings of "R" containing "E", and it is called "the subring generated by E".


title: Ring (mathematics)

text: For a ring "R", the smallest subring of "R" is called the "characteristic subring" of "R". It can be generated through addition of copies of 1 and −1. It is possible that formula_43 ("n" times) can be zero. If "n" is the smallest positive integer such that this occurs, then "n" is called the "characteristic" of "R". In some rings, formula_44 is never zero for any positive integer "n", and those rings are said to have "characteristic zero".


title: Ring (mathematics)

text: Given a ring "R", let formula_45 denote the set of all elements "x" in "R" such that "x" commutes with every element in "R": formula_46 for any "y" in "R". Then formula_45 is a subring of "R", called the center of "R". More generally, given a subset "X" of "R", let "S" be the set of all elements in "R" that commute with every element in "X". Then "S" is a subring of "R", called the centralizer (or commutant) of "X". The center is the centralizer of the entire ring "R". Elements or subsets of the center are said to be "central" in "R"; they (each individually) generate a subring of the center.


title: Ring (mathematics)

text: Let "R" be a ring. A left ideal of "R" is a nonempty subset "I" of "R" such that for any "x", "y" in "I" and "r" in "R", the elements formula_48 and formula_49 are in "I". If formula_50 denotes the "R"-span of "I", that is, the set of finite sums


title: Ring (mathematics)

text: then "I" is a left ideal if formula_52. Similarly, a right ideal is a subset "I" such that formula_53. A subset "I" is said to be a two-sided ideal or simply ideal if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of "R". If "E" is a subset of "R", then formula_54 is a left ideal, called the left ideal generated by "E"; it is the smallest left ideal containing "E". Similarly, one can consider the right ideal or the two-sided ideal generated by a subset of "R".


title: Ring (mathematics)

text: If "x" is in "R", then formula_55 and formula_56 are left ideals and right ideals, respectively; they are called the principal left ideals and right ideals generated by "x". The principal ideal formula_57 is written as formula_58. For example, the set of all positive and negative multiples of 2 along with 0 form an ideal of the integers, and this ideal is generated by the integer 2. In fact, every ideal of the ring of integers is principal.


title: Ring (mathematics)

text: Like a group, a ring is said to be simple if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field.


title: Ring (mathematics)

text: Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no strictly increasing infinite chain of left ideals is called a left Noetherian ring. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left Artinian ring. It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the Hopkins–Levitzki theorem). The integers, however, form a Noetherian ring which is not Artinian.


title: Ring (mathematics)

text: For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra. A proper ideal "P" of "R" is called a prime ideal if for any elements formula_59 we have that formula_60 implies either formula_61 or formula_62. Equivalently, "P" is prime if for any ideals formula_63 we have that formula_64 implies either formula_65 or formula_66 This latter formulation illustrates the idea of ideals as generalizations of elements.


title: Ring (mathematics)

text: A homomorphism from a ring to a ring is a function "f" from "R" to "S" that preserves the ring operations; namely, such that, for all "a", "b" in "R" the following identities hold:


title: Ring (mathematics)

text: A ring homomorphism "f" is said to be an isomorphism if there exists an inverse homomorphism to "f" (that is, a ring homomorphism that is an inverse function). Any bijective ring homomorphism is a ring isomorphism. Two rings formula_67 are said to be isomorphic if there is an isomorphism between them and in that case one writes formula_68. A ring homomorphism between the same ring is called an endomorphism and an isomorphism between the same ring an automorphism.


title: Ring (mathematics)

text: Given a ring homomorphism formula_73, the set of all elements mapped to 0 by "f" is called the kernel of "f". The kernel is a two-sided ideal of "R". The image of "f", on the other hand, is not always an ideal, but it is always a subring of "S".


title: Ring (mathematics)

text: To give a ring homomorphism from a commutative ring "R" to a ring "A" with image contained in the center of "A" is the same as to give a structure of an algebra over "R" to "A" (which in particular gives a structure of an "A"-module).


title: Ring (mathematics)

text: The notion of quotient ring is analogous to the notion of a quotient group. Given a ring and a two-sided ideal "I" of , view "I" as subgroup of ; then the quotient ring "R"/"I" is the set of cosets of "I" together with the operations


title: Ring (mathematics)

text: As with a quotient group, there is a canonical homomorphism formula_74, given by formula_75. It is surjective and satisfies the following universal property:


title: Ring (mathematics)

text: For any ring homomorphism formula_76, invoking the universal property with formula_81 produces a homomorphism formula_82 that gives an isomorphism from formula_83 to the image of .


title: Ring (mathematics)

text: The concept of a "module over a ring" generalizes the concept of a vector space (over a field) by generalizing from multiplication of vectors with elements of a field (scalar multiplication) to multiplication with elements of a ring. More precisely, given a ring with 1, an -module is an abelian group equipped with an operation (associating an element of to every pair of an element of and an element of ) that satisfies certain axioms. This operation is commonly denoted multiplicatively and called multiplication. The axioms of modules are the following: for all in and all in , we have:


title: Ring (mathematics)

text: When the ring is noncommutative these axioms define "left modules"; "right modules" are defined similarly by writing instead of . This is not only a change of notation, as the last axiom of right modules (that is ) becomes , if left multiplication (by ring elements) is used for a right module.


title: Ring (mathematics)

text: Although similarly defined, the theory of modules is much more complicated than that of vector space, mainly, because, unlike vector spaces, modules are not characterized (up to an isomorphism) by a single invariant (the dimension of a vector space). In particular, not all modules have a basis.


title: Ring (mathematics)

text: The axioms of modules imply that , where the first minus denotes the additive inverse in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.


title: Ring (mathematics)

text: Any ring homomorphism induces a structure of a module: if is a ring homomorphism, then is a left module over by the multiplication: . If is commutative or if is contained in the center of , the ring is called a -algebra. In particular, every ring is an algebra over the integers.


title: Ring (mathematics)

text: Let "R" and "S" be rings. Then the product can be equipped with the following natural ring structure:


title: Ring (mathematics)

text: for all "r", "r" in "R" and "s", "s" in "S". The ring with the above operations of addition and multiplication and the multiplicative identity formula_88 is called the direct product of "R" with "S". The same construction also works for an arbitrary family of rings: if formula_89 are rings indexed by a set "I", then formula_90 is a ring with componentwise addition and multiplication.


title: Ring (mathematics)

text: Let "R" be a commutative ring and formula_91 be ideals such that formula_92 whenever formula_93. Then the Chinese remainder theorem says there is a canonical ring isomorphism:


title: Ring (mathematics)

text: A "finite" direct product may also be viewed as a direct sum of ideals. Namely, let formula_95 be rings, formula_96 the inclusions with the images formula_97 (in particular formula_97 are rings though not subrings). Then formula_97 are ideals of "R" and


title: Ring (mathematics)

text: as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to "R". Equivalently, the above can be done through central idempotents. Assume that "R" has the above decomposition. Then we can write


title: Ring (mathematics)

text: By the conditions on formula_97, one has that formula_103 are central idempotents and formula_104 (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let formula_105, which are two-sided ideals. If each formula_103 is not a sum of orthogonal central idempotents, then their direct sum is isomorphic to "R".


title: Ring (mathematics)

text: An important application of an infinite direct product is the construction of a projective limit of rings (see below). Another application is a restricted product of a family of rings (cf. adele ring).


title: Ring (mathematics)

text: forms a commutative ring with the usual addition and multiplication, containing "R" as a subring. It is called the polynomial ring over "R". More generally, the set formula_108 of all polynomials in variables formula_109 forms a commutative ring, containing formula_110 as subrings.


title: Ring (mathematics)

text: If "R" is an integral domain, then formula_111 is also an integral domain; its field of fractions is the field of rational functions. If "R" is a Noetherian ring, then formula_111 is a Noetherian ring. If "R" is a unique factorization domain, then formula_111 is a unique factorization domain. Finally, "R" is a field if and only if formula_111 is a principal ideal domain.


title: Ring (mathematics)

text: Let formula_115 be commutative rings. Given an element "x" of "S", one can consider the ring homomorphism


title: Ring (mathematics)

text: (that is, the substitution). If and , then . Because of this, the polynomial "f" is often also denoted by formula_117. The image of the map formula_118 is denoted by formula_119; it is the same thing as the subring of "S" generated by "R" and "x".


title: Ring (mathematics)

text: Example: let "f" be a polynomial in one variable, that is, an element in a polynomial ring "R". Then formula_123 is an element in formula_124 and formula_125 is divisible by "h" in that ring. The result of substituting zero to "h" in formula_126 is formula_127, the derivative of "f" at "x".


title: Ring (mathematics)

text: The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism formula_128 and an element "x" in "S" there exists a unique ring homomorphism formula_129 such that formula_130 and formula_131 restricts to formula_132. For example, choosing a basis, a symmetric algebra satisfies the universal property and so is a polynomial ring.


title: Ring (mathematics)

text: To give an example, let "S" be the ring of all functions from "R" to itself; the addition and the multiplication are those of functions. Let "x" be the identity function. Each "r" in "R" defines a constant function, giving rise to the homomorphism formula_133. The universal property says that this map extends uniquely to


title: Ring (mathematics)

text: ("t" maps to "x") where formula_135 is the polynomial function defined by "f". The resulting map is injective if and only if "R" is infinite.


title: Ring (mathematics)

text: Given a non-constant monic polynomial "f" in formula_111, there exists a ring "S" containing "R" such that "f" is a product of linear factors in formula_137.


title: Ring (mathematics)

text: Let "k" be an algebraically closed field. The Hilbert's Nullstellensatz (theorem of zeros) states that there is a natural one-to-one correspondence between the set of all prime ideals in formula_138 and the set of closed subvarieties of formula_139. In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring. (cf. Gröbner basis.)


title: Ring (mathematics)

text: There are some other related constructions. A formal power series ring formula_140 consists of formal power series


title: Ring (mathematics)

text: together with multiplication and addition that mimic those for convergent series. It contains formula_111 as a subring. A formal power series ring does not have the universal property of a polynomial ring; a series may not converge after a substitution. The important advantage of a formal power series ring over a polynomial ring is that it is local (in fact, complete).


title: Ring (mathematics)

text: Let "R" be a ring (not necessarily commutative). The set of all square matrices of size "n" with entries in "R" forms a ring with the entry-wise addition and the usual matrix multiplication. It is called the matrix ring and is denoted by M("R"). Given a right "R"-module formula_143, the set of all "R"-linear maps from "U" to itself forms a ring with addition that is of function and multiplication that is of composition of functions; it is called the endomorphism ring of "U" and is denoted by formula_144.


title: Ring (mathematics)

text: As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring: formula_145. This is a special case of the following fact: If formula_146 is an "R"-linear map, then "f" may be written as a matrix with entries formula_147 in formula_148, resulting in the ring isomorphism:


title: Ring (mathematics)

text: Schur's lemma says that if "U" is a simple right "R"-module, then formula_144 is a division ring. If formula_151 is a direct sum of "m"-copies of simple "R"-modules formula_152, then


title: Ring (mathematics)

text: A ring "R" and the matrix ring M("R") over it are Morita equivalent: the category of right modules of "R" is equivalent to the category of right modules over M("R"). In particular, two-sided ideals in "R" correspond in one-to-one to two-sided ideals in M("R").


title: Ring (mathematics)

text: Let "R" be a sequence of rings such that "R" is a subring of "R" for all "i". Then the union (or filtered colimit) of "R" is the ring formula_154 defined as follows: it is the disjoint union of all "R"'s modulo the equivalence relation formula_155 if and only if formula_156 in "R" for sufficiently large "i".


title: Ring (mathematics)

text: A projective limit (or a filtered limit) of rings is defined as follows. Suppose we're given a family of rings formula_89, "i" running over positive integers, say, and ring homomorphisms formula_164 such that formula_165 are all the identities and formula_166 is formula_167 whenever formula_168. Then formula_169 is the subring of formula_170 consisting of formula_171 such that formula_172 maps to formula_173 under formula_164.


title: Ring (mathematics)

text: The localization generalizes the construction of the field of fractions of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ring "R" and a subset "S" of "R", there exists a ring formula_175 together with the ring homomorphism formula_176 that "inverts" "S"; that is, the homomorphism maps elements in "S" to unit elements in formula_177, and, moreover, any ring homomorphism from "R" that "inverts" "S" uniquely factors through formula_177. The ring formula_177 is called the localization of "R" with respect to "S". For example, if "R" is a commutative ring and "f" an element in "R", then the localization formula_180 consists of elements of the form formula_181 (to be precise, formula_182)


title: Ring (mathematics)

text: The localization is frequently applied to a commutative ring "R" with respect to the complement of a prime ideal (or a union of prime ideals) in "R". In that case formula_183, one often writes formula_184 for formula_177. formula_184 is then a local ring with the maximal ideal formula_187. This is the reason for the terminology "localization". The field of fractions of an integral domain "R" is the localization of "R" at the prime ideal zero. If formula_188 is a prime ideal of a commutative ring "R", then the field of fractions of formula_189 is the same as the residue field of the local ring formula_184 and is denoted by formula_191.


title: Ring (mathematics)

text: If "M" is a left "R"-module, then the localization of "M" with respect to "S" is given by a change of rings formula_192.


title: Ring (mathematics)

text: The most important properties of localization are the following: when "R" is a commutative ring and "S" a multiplicatively closed subset


title: Ring (mathematics)

text: In category theory, a localization of a category amounts to making some morphisms isomorphisms. An element in a commutative ring "R" may be thought of as an endomorphism of any "R"-module. Thus, categorically, a localization of "R" with respect to a subset "S" of "R" is a functor from the category of "R"-modules to itself that sends elements of "S" viewed as endomorphisms to automorphisms and is universal with respect to this property. (Of course, "R" then maps to formula_177 and "R"-modules map to formula_177-modules.)


title: Ring (mathematics)

text: The completion of "R" at "I" is the projective limit formula_204; it is a commutative ring. The canonical homomorphisms from "R" to the quotients formula_205 induce a homomorphism formula_206. The latter homomorphism is injective if "R" is a Noetherian integral domain and "I" is a proper ideal, or if "R" is a Noetherian local ring with maximal ideal "I", by Krull's intersection theorem. The construction is especially useful when "I" is a maximal ideal.


title: Ring (mathematics)

text: The basic example is the completion of Z at the principal ideal ("p") generated by a prime number "p"; it is called the ring of "p"-adic integers and is denoted Z. The completion can in this case be constructed also from the "p"-adic absolute value on Q. The "p"-adic absolute value on Q is a map formula_207 from Q to R given by formula_208 where formula_209 denotes the exponent of "p" in the prime factorization of a nonzero integer "n" into prime numbers (we also put formula_210 and formula_211). It defines a distance function on Q and the completion of Q as a metric space is denoted by Q. It is again a field since the field operations extend to the completion. The subring of Q consisting of elements "x" with formula_212 is isomorphic to Z.


title: Ring (mathematics)

text: Similarly, the formal power series ring formula_213 is the completion of formula_111 at formula_215 (see also Hensel's lemma)


title: Ring (mathematics)

text: A complete ring has much simpler structure than a commutative ring. This owns to the Cohen structure theorem, which says, roughly, that a complete local ring tends to look like a formal power series ring or a quotient of it. On the other hand, the interaction between the integral closure and completion has been among the most important aspects that distinguish modern commutative ring theory from the classical one developed by the likes of Noether. Pathological examples found by Nagata led to the reexamination of the roles of Noetherian rings and motivated, among other things, the definition of excellent ring.


title: Ring (mathematics)

text: The most general way to construct a ring is by specifying generators and relations. Let "F" be a free ring (that is, free algebra over the integers) with the set "X" of symbols, that is, "F" consists of polynomials with integral coefficients in noncommuting variables that are elements of "X". A free ring satisfies the universal property: any function from the set "X" to a ring "R" factors through "F" so that formula_216 is the unique ring homomorphism. Just as in the group case, every ring can be represented as a quotient of a free ring.


title: Ring (mathematics)

text: Now, we can impose relations among symbols in "X" by taking a quotient. Explicitly, if "E" is a subset of "F", then the quotient ring of "F" by the ideal generated by "E" is called the ring with generators "X" and relations "E". If we used a ring, say, "A" as a base ring instead of Z, then the resulting ring will be over "A". For example, if formula_217, then the resulting ring will be the usual polynomial ring with coefficients in "A" in variables that are elements of "X" (It is also the same thing as the symmetric algebra over "A" with symbols "X".)


title: Ring (mathematics)

text: In the category-theoretic terms, the formation formula_218 is the left adjoint functor of the forgetful functor from the category of rings to Set (and it is often called the free ring functor.)


title: Ring (mathematics)

text: Let "A", "B" be algebras over a commutative ring "R". Then the tensor product of "R"-modules formula_219 is an "R"-algebra with multiplication characterized by formula_220.


title: Ring (mathematics)

text: A nonzero ring with no nonzero zero-divisors is called a domain. A commutative domain is called an integral domain. The most important integral domains are principal ideal domains, PIDs for short, and fields. A principal ideal domain is an integral domain in which every ideal is principal. An important class of integral domains that contain a PID is a unique factorization domain (UFD), an integral domain in which every nonunit element is a product of prime elements (an element is prime if it generates a prime ideal.) The fundamental question in algebraic number theory is on the extent to which the ring of (generalized) integers in a number field, where an "ideal" admits prime factorization, fails to be a PID.


title: Ring (mathematics)

text: Among theorems concerning a PID, the most important one is the structure theorem for finitely generated modules over a principal ideal domain. The theorem may be illustrated by the following application to linear algebra. Let "V" be a finite-dimensional vector space over a field "k" and formula_221 a linear map with minimal polynomial "q". Then, since formula_122 is a unique factorization domain, "q" factors into powers of distinct irreducible polynomials (that is, prime elements):


title: Ring (mathematics)

text: Letting formula_224, we make "V" a "k"["t"]-module. The structure theorem then says "V" is a direct sum of cyclic modules, each of which is isomorphic to the module of the form formula_225. Now, if formula_226, then such a cyclic module (for formula_227) has a basis in which the restriction of "f" is represented by a Jordan matrix. Thus, if, say, "k" is algebraically closed, then all formula_227's are of the form formula_229 and the above decomposition corresponds to the Jordan canonical form of "f".


title: Ring (mathematics)

text: In algebraic geometry, UFDs arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a regular local ring. A regular local ring is a UFD.


title: Ring (mathematics)

text: The following is a chain of class inclusions that describes the relationship between rings, domains and fields:


title: Ring (mathematics)

text: A division ring is a ring such that every non-zero element is a unit. A commutative division ring is a field. A prominent example of a division ring that is not a field is the ring of quaternions. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every "finite" domain (in particular finite division ring) is a field; in particular commutative (the Wedderburn's little theorem).


title: Ring (mathematics)

text: Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field.


title: Ring (mathematics)

text: The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the Cartan–Brauer–Hua theorem.


title: Ring (mathematics)

text: A "semisimple module" is a direct sum of simple modules. A "semisimple ring" is a ring that is semisimple as a left module (or right module) over itself.


title: Ring (mathematics)

text: The Weyl algebra over a field is a simple ring, but it is not semisimple. The same holds for a ring of differential operators in many variables.


title: Ring (mathematics)

text: Any module over a semisimple ring is semisimple. (Proof: A free module over a semisimple ring is semisimple and any module is a quotient of a free module.)


title: Ring (mathematics)

text: Semisimplicity is closely related to separability. A unital associative algebra "A" over a field "k" is said to be separable if the base extension formula_231 is semisimple for every field extension formula_232. If "A" happens to be a field, then this is equivalent to the usual definition in field theory (cf. separable extension.)


title: Ring (mathematics)

text: For a field "k", a "k"-algebra is central if its center is "k" and is simple if it is a simple ring. Since the center of a simple "k"-algebra is a field, any simple "k"-algebra is a central simple algebra over its center. In this section, a central simple algebra is assumed to have finite dimension. Also, we mostly fix the base field; thus, an algebra refers to a "k"-algebra. The matrix ring of size "n" over a ring "R" will be denoted by formula_233.


title: Ring (mathematics)

text: Two central simple algebras "A" and "B" are said to be "similar" if there are integers "n" and "m" such that formula_234. Since formula_235, the similarity is an equivalence relation. The similarity classes formula_236 with the multiplication formula_237 form an abelian group called the Brauer group of "k" and is denoted by formula_238. By the Artin–Wedderburn theorem, a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring.


title: Ring (mathematics)

text: For example, formula_238 is trivial if "k" is a finite field or an algebraically closed field (more generally quasi-algebraically closed field; cf. Tsen's theorem). formula_240 has order 2 (a special case of the theorem of Frobenius). Finally, if "k" is a nonarchimedean local field (for example, formula_241), then formula_242 through the invariant map.


title: Ring (mathematics)

text: Now, if "F" is a field extension of "k", then the base extension formula_243 induces formula_244. Its kernel is denoted by formula_245. It consists of formula_236 such that formula_231 is a matrix ring over "F" (that is, "A" is split by "F".) If the extension is finite and Galois, then formula_245 is canonically isomorphic to formula_249.


title: Ring (mathematics)

text: If is a field, a valuation is a group homomorphism from the multiplicative group to a totally ordered abelian group such that, for any in with nonzero, . The valuation ring of is the subring of consisting of zero and all nonzero such that .


title: Ring (mathematics)

text: A ring may be viewed as an abelian group (by using the addition operation), with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:


title: Ring (mathematics)

text: Many different kinds of mathematical objects can be fruitfully analyzed in terms of some associated ring.


title: Ring (mathematics)

text: a graded ring. There are also homology groups formula_259 of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the spheres and tori, for which the methods of point-set topology are not well-suited. Cohomology groups were later defined in terms of homology groups in a way which is roughly analogous to the dual of a vector space. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the universal coefficient theorem. However, the advantage of the cohomology groups is that there is a natural product, which is analogous to the observation that one can multiply pointwise a "k"-multilinear form and an "l"-multilinear form to get a ()-multilinear form.


title: Ring (mathematics)

text: The ring structure in cohomology provides the foundation for characteristic classes of fiber bundles, intersection theory on manifolds and algebraic varieties, Schubert calculus and much more.


title: Ring (mathematics)

text: To any group is associated its Burnside ring which uses a ring to describe the various ways the group can act on a finite set. The Burnside ring's additive group is the free abelian group whose basis are the transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the representation ring: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.


title: Ring (mathematics)

text: To any group ring or Hopf algebra is associated its representation ring or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from character theory, which is more or less the Grothendieck group given a ring structure.


title: Ring (mathematics)

text: To any irreducible algebraic variety is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring. The study of algebraic geometry makes heavy use of commutative algebra to study geometric concepts in terms of ring-theoretic properties. Birational geometry studies maps between the subrings of the function field.


title: Ring (mathematics)

text: Every simplicial complex has an associated face ring, also called its Stanley–Reisner ring. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in algebraic combinatorics. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of simplicial polytopes.


title: Ring (mathematics)

text: Every ring can be thought of as a monoid in Ab, the category of abelian groups (thought of as a monoidal category under the tensor product of formula_260-modules). The monoid action of a ring "R" on an abelian group is simply an "R"-module. Essentially, an "R"-module is a generalization of the notion of a vector space – where rather than a vector space over a field, one has a "vector space over a ring".


title: Ring (mathematics)

text: Let be an abelian group and let End("A") be its endomorphism ring (see above). Note that, essentially, End("A") is the set of all morphisms of "A", where if "f" is in End("A"), and "g" is in End("A"), the following rules may be used to compute and :


title: Ring (mathematics)

text: where + as in is addition in "A", and function composition is denoted from right to left. Therefore, associated to any abelian group, is a ring. Conversely, given any ring, , is an abelian group. Furthermore, for every "r" in "R", right (or left) multiplication by "r" gives rise to a morphism of , by right (or left) distributivity. Let . Consider those endomorphisms of "A", that "factor through" right (or left) multiplication of "R". In other words, let End("A") be the set of all morphisms "m" of "A", having the property that . It was seen that every "r" in "R" gives rise to a morphism of "A": right multiplication by "r". It is in fact true that this association of any element of "R", to a morphism of "A", as a function from "R" to End("A"), is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian "X"-group (by "X"-group, it is meant a group with "X" being its set of operators). In essence, the most general form of a ring, is the endomorphism group of some abelian "X"-group.


title: Ring (mathematics)

text: Any ring can be seen as a preadditive category with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context. Additive functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms.


title: Ring (mathematics)

text: Algebraists have defined structures more general than rings by weakening or dropping some of ring axioms.


title: Ring (mathematics)

text: A nonassociative ring is an algebraic structure that satisfies all of the ring axioms except the associative property and the existence of a multiplicative identity. A notable example is a Lie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.


title: Ring (mathematics)

text: A semiring (sometimes "rig") is obtained by weakening the assumption that ("R", +) is an abelian group to the assumption that ("R", +) is a commutative monoid, and adding the axiom that for all "a" in "R" (since it no longer follows from the other axioms).


title: Ring (mathematics)

text: Let "C" be a category with finite products. Let pt denote a terminal object of "C" (an empty product). A ring object in "C" is an object "R" equipped with morphisms formula_262 (addition), formula_263 (multiplication), formula_264 (additive identity), formula_265 (additive inverse), and formula_266 (multiplicative identity) satisfying the usual ring axioms. Equivalently, a ring object is an object "R" equipped with a factorization of its functor of points formula_267 through the category of rings: formula_268.


title: Ring (mathematics)

text: In algebraic geometry, a ring scheme over a base scheme is a ring object in the category of -schemes. One example is the ring scheme over , which for any commutative ring returns the ring of -isotypic Witt vectors of length over .


title: Ring (mathematics)

text: In algebraic topology, a ring spectrum is a spectrum "X" together with a multiplication formula_269 and a unit map formula_270 from the sphere spectrum "S", such that the ring axiom diagrams commute up to homotopy. In practice, it is common to define a ring spectrum as a monoid object in a good category of spectra such as the category of symmetric spectra.


title: Trevor Horn

text: Trevor Charles Horn (born 15 July 1949) is an English music producer, label and recording studio owner, musician and composer. He is best known for his production work in the 1980s, and for being one half of the new wave band The Buggles (with Geoff Downes). Horn took up the bass guitar at an early age and taught himself the instrument and to sight-read music. In the 1970s, he worked as a session musician, built his own studio, and wrote and produced singles for various artists.


title: Trevor Horn

text: Horn and Downes gained international fame in 1979 with the Buggles' hit single "Video Killed the Radio Star". This was followed by their one-year tenure with the progressive rock band Yes, with Horn becoming their lead singer. In 1981, Horn became a full-time producer, working on commercially successful songs and albums for numerous artists, among them Dollar, ABC, Malcolm McLaren, Yes, and Frankie Goes to Hollywood. He ventured into business with his wife Jill Sinclair, purchasing Sarm West Studios and establishing the publishers Perfect Songs and their own label, ZTT Records. In the following year, Horn co-formed the electronic group Art of Noise. In the 1990s, Horn's success continued with his association with Seal. He has been a member of the supergroup Producers, later known as the Trevor Horn Band, since 2006.


title: Trevor Horn

text: Horn has won numerous awards, including three Brit Awards for Best British Producer in 1983, 1985, and 1992. He won a Grammy Award for producing Seal's 1994 hit "Kiss from a Rose." In 2010, Horn received an Ivor Novello Award for Outstanding Contribution to British Music. His influence on pop and electronic music in the 1980s was such that he has been called "The Man Who Invented the Eighties".


title: Trevor Horn

text: Trevor Charles Horn was born on 15 July 1949 to John and Elizabeth Horn in Hetton Le Hole, in the North East of England and grew up near the Stonebridge Pub, Durham City. The second of four children, Horn has two sisters, including novelist Marjorie DeLuca, and a brother, Ken Horn. His father was a maintenance engineer at the neighbouring dairy and a semi-professional musician who played the double bass in the Joe Clarke Big Band during the week. Horn attended Johnston Grammar School in Durham.


title: Trevor Horn

text: At around eight years of age, Horn took up the double bass and was taught the basics by his father, including the concept of playing triads. He then taught himself the bass guitar and became confident in sight-reading music, using guide books and practising on his father's four-string guitar in the spare room of the house. In his early teens, Horn would fill in for his father on the double bass in the Joe Clarke band when he was late for a gig. At school Horn was given a recorder which he picked up with little effort as he already had music knowledge, and performed in the local youth orchestra. His interests soon turned to contemporary rock acts such as the Beatles, the Rolling Stones, and Bob Dylan. At fourteen Horn played electric guitar in his first group, the Outer Limits, named after the 1963 television series of the same name, playing mainly covers by the Kinks.


title: Trevor Horn

text: Horn went on to pursue a "succession of day jobs", including one at a rubber company. He also put on a Bob Dylan imitation act for two nights a week "with a harmonica around my neck", and played the bass at odd gigs. Then, at seventeen, Horn decided to pursue a career in music and "woke my parents up at 4am to tell them". They were reluctant at first as they wanted him to become a chartered accountant as he performed well in maths, but Horn had failed the required exams. Horn's parents pleaded with him to try one more job, but three months into his role as a progress chaser in a plastic bag factory, he was fired. "I said, 'That's it, I'm never going into that world again!'", and the next day, received an offer to play the bass in a local semi-professional band at a Top Rank Ballroom, playing top 40 and dance music for £24 a week for five nights' work. Horn also received airplay on BBC Radio Leicester, performing self-written songs on a guitar.


title: Trevor Horn

text: At 21, Horn relocated to London and took up work by playing in a band which involved re-recording top 20 songs for BBC radio due to the needle time restrictions then in place. This was followed by a one-year tenure with Ray McVay's big band, and earning as a session musician, producing jingles, records, and working with rock groups. His time with Ray McVay included performances at the world ballroom dancing championship and the television show "Come Dancing". At 24, Horn began a period of work in Leicester where his growing interest in recording studios led to his assistance in the construction of a studio in the city, while playing the bass at Bailey's club for seven nights for money. Upon completion of the studio, Horn produced songs for local artists, including a song for Leicester City F.C.


title: Trevor Horn

text: By 1976, Horn had returned to London. He played bass in Northern Lights, a covers band, which also featured keyboardist Geoff Downes and disco singer Tina Charles. Horn went on to form Tracks, a jazz fusion band inspired by Weather Report and Herbie Hancock, with future Shakatak drummer Roger Odell, before he left the group to play in Tina Charles's backing band. The two entered a short relationship, and Horn learned a lot from her inspiring producer Biddu. Also featured in Charles's band were Horn's future Buggles partners, keyboardist Geoffrey Downes and guitarist Bruce Woolley.


title: Trevor Horn

text: In the mid-1970s, Horn worked for a music publishers on Denmark Street in London, producing demos which sometimes had Horn in charge of producing a master if a deal for a demo was made. From 1977 to 1979, Horn worked on various singles either as a songwriter, producer, or orchestra director, "without ever making any money out of it". Among his first was "Natural Dance" by Tony Cole and "Don't Come Back" by Fallen Angel and the T.C. Band, featuring Woolley as songwriter, which Horn produced under the name "T.C. Horn". He wrote "Boot Boot Woman", the B-side to the Boogatti single "Come Back Marianne", under his real name. In 1978, Horn wrote, sang, and produced "Caribbean Air Control" under the pseudonym Big A, which features Horn pictured as a pilot on the front sleeve. In 1979, a full studio album by Chromium, a "sci-fi disco project" named "Star to Star" was released that featured Horn and Downes as songwriters and producers, and Horn's future Art of Noise bandmate Anne Dudley on keyboards. Other artists that Horn worked with included Woolley, John Howard, Dusty Springfield ("Baby Blue"), and the Jags ("Back of My Hand"). Horn scored his first production hit when "Monkey Chop" by Dan-I reached No. 30 on the UK singles chart in 1979.


title: Trevor Horn

text: In 1978, Horn and Downes formed the new wave band the Buggles with early contributions from Woolley. They secured a recording deal with Island Records and spent much of 1979 recording their debut album, "The Age of Plastic" (1980). The credits list Horn with co-production, lead vocals, guitar and bass. Its lead single "Video Killed the Radio Star" was released in September 1979 and reached No. 1 in the UK, propelling Horn, then aged 30, and Downes to mainstream fame. In August 1981, the song was the first music video to air on MTV.


title: Trevor Horn

text: The success of "Video Killed the Radio Star" led to Horn and Downes to secure management from Brian Lane, who was also managing the progressive rock band Yes. They were in need of a singer and keyboardist following the departures of Jon Anderson and Rick Wakeman, which led to Horn and Downes pitching "We Can Fly from Here", a demo that they had written with Yes in mind. Both accepted to join Yes and work got underway on "Drama" (1980) with Horn on lead vocals and fretless bass. Horn spent much of his time on the album, and cut his wedding reception short in order to resume working on it. Horn sang on the band's 1980 tour of North America and the UK, after which he left to become a full-time producer.


title: Trevor Horn

text: In 1981, he completed a second Buggles album "Adventures in Modern Recording" largely on his own following Downes's decision to form Asia. Horn resumed working with Yes as a producer on their albums "90125" (1983) and "Big Generator" (1987). Horn rated "Owner of a Lonely Heart", the lead single from "90125", as technically his best work.


title: Trevor Horn

text: In early 1981, Horn left Yes and became a full-time producer. His wife advised him to branch off from being an instrumental musician as he could reach greater success in production, and subsequently she became Horn's manager. He assembled a studio rig which included a Roland TR-808 drum machine and sequencer and a set of Simmons electronic drum modules. He spent £18,000 on a Fairlight CMI synthesiser, one of four in the country at the time. "I knew what it was capable of, because I understood what it did. Most other people didn't understand at the time – sampling was like a mystical world". Horn realised he needed full-time assistance in operating the machine and hired J. J. Jeczalik to programme it. For his use of the Fairlight, Horn is credited as the "key architect" in incorporating sampling into "the language of pop".


title: Trevor Horn

text: Horn had commercial success with his first project, "The Dollar Album" (1982) by pop duo Dollar, which his wife had assigned him to work on. He co-wrote and produced four songs that follow a love story across them: "Mirror Mirror", "Hand Held in Black and White", "Give Me Back My Heart", and "Videotheque". All four became top 20 hits in the UK. Horn's production style attracted interest from other bands, leading to even greater success with "The Lexicon of Love" (1982) by ABC, which reached No. 1 on the UK albums chart. It was during these sessions that Horn acquired a LinnDrum drum machine, and assembled a team that would characterise and define the sound of much of his work in the 1980s, with Dudley on keyboards and arrangements, Gary Langan and later Stephen Lipson as chief engineer, Jeczalik as programmer, backing vocalist Tessa Webb, and percussionist Luis Jardim.


title: Trevor Horn

text: In 1982, Horn and his wife formed a music publishing company, Perfect Songs. This coincided with their then recent acquisition of Basing Street Studios, which also housed the fledgling publishing company. Perfect Songs was able to harness and develop the up-and-coming young artists working in the recording studio. The first to be signed were Frankie Goes to Hollywood, followed by Art of Noise and Propaganda. These first few signings to the company were instrumental in establishing the company ethos of "innovation and artiste development, taking risks and signing acts far into the left field". In 1983, Horn and his wife purchased Basing Street Studios in west London from Chris Blackwell and renamed it Sarm West Studios. The deal included the rights for them to operate a record label through Island Record's distribution which led to the formation of ZTT Records with "NME" writer Paul Morley. It is named after the sound poem of the same name by Italian futurist Filippo Tommaso Marinetti.


title: Trevor Horn

text: During 1982 and 1983, Horn worked with Malcolm McLaren and Anne Dudley, writing numerous worldwide hits including "Buffalo Gals", "Double Dutch", "Duck for the Oyster" and the "Duck Rock" album.


title: Trevor Horn

text: In 1983, Horn also co-formed Art of Noise, co-writing several hits including "Close (To the Edit)", "Beat Box", "Moments in Love", and "Slave to the Rhythm". This was originally intended as Frankie Goes to Hollywood's second single, but was instead given to Grace Jones. Horn and his studio team reworked and reinterpreted it, jazz style, into six separate songs to form Jones's album "Slave to the Rhythm". Horn got David Gilmour to play the guitar.


title: Trevor Horn

text: In 1984, Horn was approached by Bob Geldof to produce the song "Do They Know It's Christmas?", but he was unavailable. Instead, he gave use of Sarm West Studio free of charge to the project for 24 hours, which Geldof accepted, assigning Midge Ure as the producer instead. The song was recorded and mixed on 25 November. Horn produced the B-side featuring messages from artists who had and had not made the recording, including David Bowie, Annie Lennox, Paul McCartney, Big Country, and Holly Johnson. They were also recorded over the same backing track as the "Do They Know It's Christmas?"


title: Trevor Horn

text: Several musicians have described Horn's style of production as dominating. Frankie Goes to Hollywood's debut album "Welcome to the Pleasuredome" barely featured any of the band's performances, instead featuring Horn and session musicians (lead single "Relax" cost £70,000 spent across three sessions that included scrapped versions by the band and by Ian Dury's backing band, before Horn re-recorded the song himself); and the Pet Shop Boys remarked that although Horn had promised to complete their single "Left to My Own Devices" in a couple of weeks, it took several months for them to receive the final mix due to the lavish live orchestration and studio work.


title: Trevor Horn

text: In the late 1980s, Horn relocated to Bel Air, Los Angeles where he established Sarm West Coast LA, a residential recording studio.