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
Canonical name	PrincipalIdealRing
Date of creation	2013-03-22 14:33:16
Last modified on	2013-03-22 14:33:16
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	7
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 13F10
Classification	msc 13A15
Synonym	principal ring
Related topic	CriterionForCyclicRingsToBePrincipalIdealRings