What is "principal ideal ring"

n. 1 (context algebra ring theory English) A commutative ring in which every ideal is a principal ideal. 2 (context algebra ring theory English) A noncommutative ring in which every left ideal is a principal ideal and so is every right ideal.

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.