zero divisor ZeroDivisor 2002-07-06 14:23:00 2006-11-01 13:35:42 Definition left zero divisor right zero divisor regular element % 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 Let $a$ be a nonzero element of a ring $R$. The element $a$ is a {\em left zero divisor} if there exists a nonzero element $b \in R$ such that $a \cdot b = 0$. Similarly, $a$ is a {\em right zero divisor} if there exists a nonzero element $c \in R$ such that $c \cdot a = 0$. The element $a$ is said to be a {\em zero divisor} if it is both a left and right zero divisor. A nonzero element $a \in R$ is said to be a {\em regular element} if it is neither a left nor a right zero divisor. {\bf Example:} Let $R = \mathbb{Z}_6$. Then the elements $2$ and $3$ are zero divisors, since $2 \cdot 3 \equiv 6 \equiv 0 \pmod 6$.