# Difference between revisions of "Zero divisor"

{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In abstract algebra, an element Template:Mvar of a ring Template:Mvar is called a left zero divisor if there exists a nonzero Template:Mvar such that ax = 0, or equivalently if the map from Template:Mvar to Template:Mvar that sends Template:Mvar to Template:Mvar is not injective.[1] Similarly, an element Template:Mvar of a ring is called a right zero divisor if there exists a nonzero Template:Mvar such that ya = 0. This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor.[2] An element Template:Mvar that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero Template:Mvar such that ax = 0 may be different from the nonzero Template:Mvar such that ya = 0). If the ring is commutative, then the left and right zero divisors are the same.

An element of a ring that is not a zero divisor is called regular, or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor.

## Properties

• Left or right zero divisors can never be units, because if Template:Mvar is invertible and ax = 0, then 0 = a−10 = a−1ax = x, whereas x must be nonzero.

## Zero as a zero divisor

There is no need for a separate convention regarding the case a = 0, because the definition applies also in this case:

• If Template:Mvar is a ring other than the zero ring, then 0 is a (two-sided) zero divisor, because 0 · 1 = 0 and 1 · 0 = 0.
• If Template:Mvar is the zero ring, in which 0 = 1, then 0 is not a zero divisor, because there is no nonzero element that when multiplied by 0 yields 0.

Such properties are needed in order to make the following general statements true:

Some references choose to exclude 0 as a zero divisor by convention, but then they must introduce exceptions in the two general statements just made.

## Zero divisor on a module

Let Template:Mvar be a commutative ring, let Template:Mvar be an Template:Mvar-module, and let Template:Mvar be an element of Template:Mvar. One says that Template:Mvar is Template:Mvar-regular if the multiplication by Template:Mvar map ${\displaystyle M{\stackrel {a}{\to }}M}$ is injective, and that Template:Mvar is a zero divisor on Template:Mvar otherwise.[3] The set of Template:Mvar-regular elements is a multiplicative set in Template:Mvar.[4]

Specializing the definitions of "Template:Mvar-regular" and "zero divisor on Template:Mvar" to the case Template:Mvar = Template:Mvar recovers the definitions of "regular" and "zero divisor" given earlier in this article.

## Notes

1. See Bourbaki, p. 98.
2. See Lanski (2005).
3. Matsumura, p. 12
4. Matsumura, p. 12

## References

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