Principal ideal ring

In mathematics, a principal right (left) ideal ring is a ring "R" in which every right (left) ideal is of the form "xR" ("Rx") for some element "x" of "R". (The right and left ideals of this form, generated by one element, are called principal ideals.) When this is satisfied for both left and right ideals, such as the case when "R" is a commutative ring, "R" can be called a principal ideal ring, or simply principal ring.
If only the finitely generated right ideals of "R" are principal, then "R" is called a right Bézout ring. Left Bézout rings are defined similarly. These conditions are studied in domains as Bézout domains.
A commutative principal ideal ring which is also an integral domain is said to be a "principal ideal domain" (PID). In this article the focus is on the more general concept of a principal ideal ring which is not necessarily a domain.
General properties.
If "R" is a right principal ideal ring, then it is certainly a right Noetherian ring, since every right ideal is finitely generated. It is also a right Bézout ring since all finitely generated right ideals are principal. Indeed, it is clear that principal right ideal rings are exactly the rings which are both right Bézout and right Noetherian.
Principal right ideal rings are closed under finite direct products. If formula_1, then each right ideal of "R" is of the form formula_2, where each formula_3 is a right ideal of "R"i. If all the "R"i are principal right ideal rings, then "A"i="x"i"R"i, and then it can be seen that formula_4. Without much more effort, it can be shown that right Bézout rings are also closed under finite direct products.
Principal right ideal rings and right Bézout rings are also closed under quotients, that is, if "I" is a proper ideal of principal right ideal ring "R", then the quotient ring "R/I" is also principal right ideal ring. This follows readily from the isomorphism theorems for rings.
All properties above have left analogues as well.
Commutative examples.
1. The ring of integers: formula_5
2. The integers modulo "n": formula_6.
3. Let formula_7 be rings and formula_8. Then "R" is a principal ring if and only if "R""i" is a principal ring for all "i".
4. The localization of a principal ring at any multiplicative subset is again a principal ring. Similarly, any quotient of a principal ring is again a principal ring.
5. Let "R" be a Dedekind domain and "I" be a nonzero ideal of "R". Then the quotient "R"/"I" is a principal ring. Indeed, we may factor "I" as a product of prime
powers: formula_9, and by the Chinese Remainder Theorem
formula_10, so it suffices to see that each
formula_11 is a principal ring. But formula_11 is isomorphic to the quotient formula_13 of the discrete valuation ring
formula_14 and, being a quotient of a principal ring, is itself a principal ring.
6. Let "k" be a finite field and put formula_15, formula_16 and formula_17. Then R is a finite local ring which is "not" principal.
7. Let "X" be a finite set. Then formula_18 forms a commutative principal ideal ring with unity, where formula_19 represents set symmetric difference and formula_20 represents the powerset of "X". If "X" has at least two elements, then the ring also has zero divisors. If "I" is an ideal, then formula_21. If instead "X" is infinite, the ring is "not" principal: take the ideal generated by the finite subsets of "X", for example.
Structure theory for commutative PIR's.
The principal rings constructed in Example 5. above are always Artinian rings; in particular they are isomorphic to a finite direct product of principal Artinian local rings.
A local Artinian principal ring is called a special principal ring and has an extremely simple ideal structure: there are only finitely many ideals, each of which is a power of the maximal ideal. For this reason, special principal rings are examples of uniserial rings.
The following result gives a complete classification of principal rings in terms of special principal rings and principal ideal domains.
Zariski–Samuel theorem: Let "R" be a principal ring. Then "R" can be written as a direct product formula_22, where each "R"i is either a principal ideal domain or a special principal ring.
The proof applies the Chinese Remainder theorem to a minimal primary decomposition of the zero ideal.
There is also the following result, due to Hungerford:
Theorem (Hungerford): Let "R" be a principal ring. Then "R" can be written as a direct product formula_22, where each "R"i is a quotient of a principal ideal domain.
The proof of Hungerford's theorem employs Cohen's structure theorems for complete local rings.
Arguing as in Example 3. above and using the Zariski-Samuel theorem, it is easy to check that Hungerford's theorem is equivalent to the statement that any special principal ring is the quotient of a discrete valuation ring.
Noncommutative examples.
Every semisimple ring "R" which is not just a product of fields is a noncommutative right and left principal ideal domain. Every right and left ideal is a direct summand of "R", and so is of the form "eR" or "Re" where "e" is an idempotent of "R". Paralleling this example, von Neumann regular rings are seen to be both right and left Bézout rings.
If "D" is a division ring and formula_24 is a ring endomorphism which is not an automorphism, then the skew polynomial ring formula_25 is known to be a principal left ideal domain which is not right Noetherian, and hence it cannot be a principal right ideal ring. This shows that even for domains principal left and principal right ideal rings are different. 