Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any other integer results in another even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group.
Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring. For instance, the prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized to ideals. There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory).
The related, but distinct, concept of an ideal in order theory is derived from the notion of ideal in ring theory. A fractional ideal is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity.
History.
Ernst Kummer invented the concept of ideal numbers to serve as the "missing" factors in number rings in which unique factorization fails; here the word "ideal" is in the sense of existing in imagination only, in analogy with "ideal" objects in geometry such as points at infinity.
In 1876, Richard Dedekind replaced Kummer's undefined concept by concrete sets of numbers, sets that he called ideals, in the third edition of Dirichlet's book "Vorlesungen über Zahlentheorie", to which Dedekind had added many supplements.
Later the notion was extended beyond number rings to the setting of polynomial rings and other commutative rings by David Hilbert and especially Emmy Noether.
Definitions and motivation.
For an arbitrary ring formula_1, let formula_2 be its additive group. A subset formula_3 is called a left ideal of formula_4 if it is an additive subgroup of formula_4 that "absorbs multiplication from the left by elements of formula_4"; that is, formula_3 is a left ideal if it satisfies the following two conditions:
A right ideal is defined with the condition replaced by . A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. In the language of modules, the definitions mean that a left (resp. right, two-sided) ideal of "R" is precisely a left (resp. right, bi-) "R"-submodule of "R" when "R" is viewed as an "R"-module. When "R" is a commutative ring, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.
To understand the concept of an ideal, consider how ideals arise in the construction of rings of "elements modulo". For concreteness, let us look at the ring ℤ"n" of integers modulo a given integer "n" ∈ ℤ (note that ℤ is a commutative ring). The key observation here is that we obtain ℤ"n" by taking the integer line ℤ and wrapping it around itself so that various integers get identified. In doing so, we must satisfy two requirements: 1) "n" must be identified with 0 since "n" is congruent to 0 modulo "n", and 2) the resulting structure must again be a ring. The second requirement forces us to make additional identifications (i.e., it determines the precise way in which we must wrap ℤ around itself). The notion of an ideal arises when we ask the question: What is the exact set of integers that we are forced to identify with 0? The answer is, unsurprisingly, the set of all integers congruent to 0 modulo "n". That is, we must wrap ℤ around itself infinitely many times so that the integers ..., , , , , ... will all align with 0. If we look at what properties this set must satisfy in order to ensure that ℤ"n" is a ring, then we arrive at the definition of an ideal. Indeed, one can directly verify that "n"ℤ is an ideal of ℤ.
Remark. Identifications with elements other than 0 also need to be made. For example, the elements in must be identified with 1, the elements in must be identified with 2, and so on. Those, however, are uniquely determined by "n"ℤ since ℤ is an additive group.
We can make a similar construction in any commutative ring "R": start with an arbitrary , and then identify with 0 all elements of the ideal It turns out that the ideal "xR" is the smallest ideal that contains "x", called the ideal generated by "x". More generally, we can start with an arbitrary subset , and then identify with 0 all the elements in the ideal generated by "S": the smallest ideal ("S") such that . The ring that we obtain after the identification depends only on the ideal ("S") and not on the set "S" that we started with. That is, if , then the resulting rings will be the same.
Therefore, an ideal "I" of a commutative ring "R" captures canonically the information needed to obtain the ring of elements of "R" modulo a given subset . The elements of "I", by definition, are those that are congruent to zero, that is, identified with zero in the resulting ring. The resulting ring is called the quotient of "R" by "I" and is denoted "R"/"I". Intuitively, the definition of an ideal postulates two natural conditions necessary for "I" to contain all elements designated as "zeros" by "R"/"I":
It turns out that the above conditions are also sufficient for "I" to contain all the necessary "zeros": no other elements have to be designated as "zero" in order to form "R"/"I". (In fact, no other elements should be designated as "zero" if we want to make the fewest identifications.)
Remark. The above construction still works using two-sided ideals even if "R" is not necessarily commutative.
Examples and properties.
"For the sake of brevity, some results are stated only for left ideals but are usually also true for right ideals with appropriate notation changes."
Types of ideals.
"To simplify the description all rings are assumed to be commutative. The non-commutative case is discussed in detail in the respective articles."
Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings. Different types of ideals are studied because they can be used to construct different types of factor rings.
Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details:
Ideal operations.
The sum and product of ideals are defined as follows. For formula_18 and formula_62, left (resp. right) ideals of a ring "R", their sum is
which is a left (resp. right) ideal,
and, if formula_59 are two-sided,
i.e. the product is the ideal generated by all products of the form "ab" with "a" in formula_18 and "b" in formula_62.
Note formula_99 is the smallest left (resp. right) ideal containing both formula_18 and formula_62 (or the union formula_102), while the product formula_103 is contained in the intersection of formula_18 and formula_62.
The distributive law holds for two-sided ideals formula_106,
If a product is replaced by an intersection, a partial distributive law holds:
where the equality holds if formula_18 contains formula_62 or formula_112.
Remark: The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given ring forms a complete modular lattice. The lattice is not, in general, a distributive lattice. The three operations of intersection, sum (or join), and product make the set of ideals of a commutative ring into a quantale.
If formula_59 are ideals of a commutative ring "R", then formula_114 in the following two cases (at least) 
An integral domain is called a Dedekind domain if for each pair of ideals formula_119, there is an ideal formula_112 such that formula_121. It can then be shown that every nonzero ideal of a Dedekind domain can be uniquely written as a product of maximal ideals, a generalization of the fundamental theorem of arithmetic.
Examples of ideal operations.
In formula_32 we have
since formula_124 is the set of integers which are divisible by both formula_125 and formula_126.
Let formula_127 and let formula_128. Then,
In the first computation, we see the general pattern for taking the sum of two finitely generated ideals, it is the ideal generated by the union of their generators. In the last three we observe that products and intersections agree whenever the two ideals intersect in the zero ideal. These computations can be checked using Macaulay2.
Radical of a ring.
Ideals appear naturally in the study of modules, especially in the form of a radical.
Let "R" be a commutative ring. By definition, a primitive ideal of "R" is the annihilator of a (nonzero) simple "R"-module. The Jacobson radical formula_135 of "R" is the intersection of all primitive ideals. Equivalently,
Indeed, if formula_137 is a simple module and "x" is a nonzero element in "M", then formula_138 and formula_139, meaning formula_140 is a maximal ideal. Conversely, if formula_141 is a maximal ideal, then formula_141 is the annihilator of the simple "R"-module formula_143. There is also another characterization (the proof is not hard):
For a not-necessarily-commutative ring, it is a general fact that formula_145 is a unit element if and only if formula_146 is (see the link) and so this last characterization shows that the radical can be defined both in terms of left and right primitive ideals.
The following simple but important fact (Nakayama's lemma) is built-in to the definition of a Jacobson radical: if "M" is a module such that formula_147, then "M" does not admit a maximal submodule, since if there is a maximal submodule formula_148, formula_149 and so formula_150, a contradiction. Since a nonzero finitely generated module admits a maximal submodule, in particular, one has:
A maximal ideal is a prime ideal and so one has
where the intersection on the left is called the nilradical of "R". As it turns out, formula_154 is also the set of nilpotent elements of "R".
If "R" is an Artinian ring, then formula_155 is nilpotent and formula_156. (Proof: first note the DCC implies formula_157 for some "n". If (DCC) formula_158 is an ideal properly minimal over the latter, then formula_159. That is, formula_160, a contradiction.)
Extension and contraction of an ideal.
Let "A" and "B" be two commutative rings, and let "f" : "A" → "B" be a ring homomorphism. If formula_18 is an ideal in "A", then formula_162 need not be an ideal in "B" (e.g. take "f" to be the inclusion of the ring of integers Z into the field of rationals Q). The extension formula_163 of formula_18 in "B" is defined to be the ideal in "B" generated by formula_162. Explicitly,
If formula_62 is an ideal of "B", then formula_168 is always an ideal of "A", called the contraction formula_169 of formula_62 to "A".
Assuming "f" : "A" → "B" is a ring homomorphism, formula_18 is an ideal in "A", formula_62 is an ideal in "B", then:
It is false, in general, that formula_18 being prime (or maximal) in "A" implies that formula_163 is prime (or maximal) in "B". Many classic examples of this stem from algebraic number theory. For example, embedding formula_180. In formula_181, the element 2 factors as formula_182 where (one can show) neither of formula_183 are units in "B". So formula_184 is not prime in "B" (and therefore not maximal, as well). Indeed, formula_185 shows that formula_186, formula_187, and therefore formula_188.
On the other hand, if "f" is surjective and formula_189 then:
Remark: Let "K" be a field extension of "L", and let "B" and "A" be the rings of integers of "K" and "L", respectively. Then "B" is an integral extension of "A", and we let "f" be the inclusion map from "A" to "B". The behaviour of a prime ideal formula_198 of "A" under extension is one of the central problems of algebraic number theory.
The following is sometimes useful: a prime ideal formula_199 is a contraction of a prime ideal if and only if formula_200. (Proof: Assuming the latter, note formula_201 intersects formula_202, a contradiction. Now, the prime ideals of formula_203 correspond to those in "B" that are disjoint from formula_202. Hence, there is a prime ideal formula_205 of "B", disjoint from formula_202, such that formula_207 is a maximal ideal containing formula_208. One then checks that formula_205 lies over formula_199. The converse is obvious.)
Generalisations.
Ideals can be generalised to any monoid object formula_211, where formula_4 is the object where the monoid structure has been forgotten. A left ideal of formula_4 is a subobject formula_3 that "absorbs multiplication from the left by elements of formula_4"; that is, formula_3 is a left ideal if it satisfies the following two conditions:
A right ideal is defined with the condition "formula_223" replaced by "'formula_224". A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. When formula_4 is a commutative monoid object respectively, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.
An ideal can also be thought of as a specific type of -module. If we consider formula_4 as a left formula_4-module (by left multiplication), then a left ideal formula_3 is really just a left sub-module of formula_4. In other words, formula_3 is a left (right) ideal of formula_4 if and only if it is a left (right) formula_4-module which is a subset of formula_4. formula_3 is a two-sided ideal if it is a sub-formula_4-bimodule of formula_4.
Example: If we let formula_237, an ideal of formula_32 is an abelian group which is a subset of formula_32, i.e. formula_240 for some formula_241. So these give all the ideals of formula_32.