# principal ideal ring

A commutative ring $R$ in which all ideals are principal (http://planetmath.org/PrincipalIdeal), i.e. (http://planetmath.org/Ie) generated by (http://planetmath.org/IdealGeneratedBy) a single ring element, is called a principal ideal ring.  If $R$ is also an integral domain, it is a principal ideal domain.

Some well-known principal ideal rings are the ring $\mathbb{Z}$ of integers, its factor rings $\mathbb{Z}/n\mathbb{Z}$, and any polynomial ring over a field.

Title principal ideal ring PrincipalIdealRing 2013-03-22 14:33:16 2013-03-22 14:33:16 Wkbj79 (1863) Wkbj79 (1863) 7 Wkbj79 (1863) Definition msc 13F10 msc 13A15 principal ring CriterionForCyclicRingsToBePrincipalIdealRings