Associative rings and algebras Definition

Zero divisor

Builds on Ring

Definition

Let $R$ be a ring. An element $a \in R$ is a left zero divisor if there is a nonzero $x \in R$ with $a x = 0$ . An element $a \in R$ is a right zero divisor if there is a nonzero $y \in R$ with $y a = 0$ . An element is a zero divisor if it is a left or a right zero divisor.

Remarks

In a commutative ring, left and right zero divisors are the same. In $\mathbb{Z}/6\mathbb{Z}$ , the classes of $2$ and $3$ are zero divisors, because $2 \cdot 3 = 0$ . Some authors do not count $0$ as a zero divisor. With the definition above, $0$ is a zero divisor of every nonzero ring. A unit is never a zero divisor.

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Group concept
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Zero divisor definition

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Tag	`0K26YKNBSA`
URI	`m2://0K26YKNBSA`
Kind	definition
Writing	W1 stated
MSC	16U99
Defines	Zero divisor
Uses	Ring

Zero divisor

Definition

Remarks

Reading path

Where this leads

Further reading

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