prime element PrimeElement 2002-06-12 00:48:42 2004-04-24 20:18:49 Definition % 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 here An element $p$ in a ring $R$ is a prime element if it generates a prime ideal. If $R$ is commutative, this is equivalent to saying that for all $a,b \in R$ , if $p$ divides $ab$, then $p$ divides $a$ or $p$ divides $b$. \\ When $R = \mathbb{Z}$ the prime elements as formulated above are simply prime numbers.