PrimalElement 2004-11-22 00:25:37 2008-08-21 22:48:51 Definition primal % 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,amscd} \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 $r$ in a commutative ring $R$ is called \emph{primal} if whenever $r\mid ab$, with $a,b\in R$, then there exist elements $s,t\in R$ such that \begin{enumerate} \item $r=st$, \item $s\mid a$ and $t\mid b$. \end{enumerate} \textbf{Lemma}. In a commutative ring, an element that is both irreducible and primal is a prime element. \begin{proof} Suppose $a$ is irreducible and primal, and $a\mid bc$. Since $a$ is primal, there is $x,y\in R$ such that $a=xy$, with $x\mid b$ and $y\mid c$. Since $a$ is irreducible, either $x$ or $y$ is a unit. If $x$ is a unit, with $z$ as its inverse, then $za=zxy=y$, so that $a\mid y$. But $y\mid c$, we have that $a\mid c$. \end{proof}