principal ideal ringPrincipalIdealRing2004-08-23 17:38:552007-05-30 16:43:18Definition% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands hereA commutative ring $R$ in which all ideals are \PMlinkname{principal}{PrincipalIdeal}, \PMlinkname{i.e.}{Ie} \PMlinkname{generated by}{IdealGeneratedBy} a single ring element, is called a {\em 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.