# Zero divisor

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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**.

## Examples

- In the ring , the residue class is a zero divisor since .
- The ring of integers has no zero divisors except for 0.
- A nilpotent element of a nonzero ring is always a two-sided zero divisor.
- A idempotent element of a ring is always a two-sided zero divisor, since .
- Examples of zero divisors in the ring of matrices (over any nonzero ring) are shown here:
- A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in
*R*_{1}×*R*_{2}with each*R*_{i}nonzero, (1,0)(0,1) = (0,0), so (1,0) is a zero divisor.

### One-sided zero-divisor

- Consider the ring of (formal) matrices with and . Then and . If , then is a left zero divisor iff is even, since ; and it is a right zero divisor iff is even for similar reasons. If either of is , then it is a two-sided zero-divisor.
- Here is another example of a ring with an element that is a zero divisor on one side only. Let be the set of all sequences of integers . Take for the ring all additive maps from to , with pointwise addition and composition as the ring operations. (That is, our ring is , the
**endomorphism ring**of the additive group .) Three examples of elements of this ring are the**right shift**, the**left shift**, and the**projection map**onto the first factor . All three of these additive maps are not zero, and the composites and are both zero, so is a left zero divisor and is a right zero divisor in the ring of additive maps from to . However, is not a right zero divisor and is not a left zero divisor: the composite is the identity. Note also that is a two-sided zero-divisor since , while is not in any direction.

## Non-examples

- The ring of integers modulo a prime number has no zero divisors except 0. Since every nonzero element is a unit, this ring is a field.

- More generally, a division ring has no zero divisors except 0.

- A nonzero commutative ring whose only zero divisor is 0 is called an integral domain.

## Properties

- In the ring of Template:Mvar-by-Template:Mvar matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of Template:Mvar-by-Template:Mvar matrices over an integral domain, the zero divisors are precisely the matrices with determinant zero.

- Left or right zero divisors can never be units, because if Template:Mvar is invertible and
*ax*= 0, then 0 =*a*^{−1}0 =*a*^{−1}*ax*=*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:

- In a commutative ring Template:Mvar, the set of non-zero-divisors is a multiplicative set in Template:Mvar. (This, in turn, is important for the definition of the total quotient ring.) The same is true of the set of non-left-zero-divisors and the set of non-right-zero-divisors in an arbitrary ring, commutative or not.
- In a commutative ring Template:Mvar, the set of zero divisors is the union of the associated prime ideals of Template:Mvar.

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 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.

## See also

- Zero-product property
- Glossary of commutative algebra (Exact zero divisor)

## Notes

## References

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