principal ideal ring

For some authors (e.g., Bourbaki, 1964), the same as principal ideal domain. Most authors, however, do not require the ring to be an integral domain, and define a principal ring (sometimes also called a principal ideal ring) simply as a commutative unit ring (different from the zero ring) in which every ideal is principal, i.e., can be generated by a single element. Examples include the ring of integers Z, any field, and any polynomial ring in one variable over a field. While all Euclidean rings are principal rings, the converse is not true.

Euclidean ring | irreducible element | prime element | principal ideal