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# 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 $2\cdot 3\equiv 6\equiv 0\;\;(\mathop{{\rm mod}}6)$.

Defines:

left zero divisor, right zero divisor, regular element

Related:

CancellationRing, IntegralDomain, Unity

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

13G05*no label found*

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## Comments

## regular element

There are now possibly 2 different definitions of a regular element of an element of a ring and they are not equivalent. First an element that is neither a left zero divisor nor right zero divisor and secondly p is regular in a ring if there exists s in R such that p = psp.

## Re: regular element

There are several mathematicians in whose books the second definition of regular is used namely : I. Herstein in Noncommutative Rings and N. McCoy in the Theory of Rings.

## Re: regular element

Change the comment about matjematicians to N. McCoy in the Theory of Rings.

## Re: regular element

Change the comment about mathematicians to N. McCoy in the Theory of Rings and Seth Warner in Modern Algebra about regular rings.