zero divisor

Let $a$ be a nonzero element of a ring $R$.

The element $a$ is a left zero divisor if there exists a nonzero element $b\in R$ such that $a\cdot b=0$. Similarly, $a$ is a 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 zero divisor if it is both a left and right zero divisor. A nonzero element $a\in R$ is said to be a regular element if it is neither a left nor a right zero divisor.

Example: Let $R={\mathbb{Z}}_6$. Then the elements $2$ and $3$ are zero divisors, since $2\cdot 3\equiv 6\equiv 0(mod6)$.

Title	zero divisor
Canonical name	ZeroDivisor
Date of creation	2013-03-22 12:49:59
Last modified on	2013-03-22 12:49:59
Owner	cvalente (11260)
Last modified by	cvalente (11260)
Numerical id	9
Author	cvalente (11260)
Entry type	Definition
Classification	msc 13G05
Related topic	CancellationRing
Related topic	IntegralDomain
Related topic	Unity
Defines	left zero divisor
Defines	right zero divisor
Defines	regular element