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Concept
Principal ideal ring
Summary
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.
If R is a principal right 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 , then each right ideal of R is of the form , where each is a right ideal of Ri. If all the Ri are principal right ideal rings, then Ai=xiRi, and then it can be seen that . 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.
- The ring of integers:
- The integers modulo n: .
- Let be rings and . Then R is a principal ring if and only if Ri is a principal ring for all i.
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