# Zero divisor

In abstract algebra, an element a of a ring R is called a **left zero divisor** if there exists a nonzero x such that *ax* = 0,^{[1]} or equivalently if the map from *R* to *R* that sends x to ax is not injective.^{[2]} Similarly, an element a of a ring is called a **right zero divisor** if there exists a nonzero y 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**.^{[3]} An element a that is both a left and a right zero divisor is called a **two-sided zero divisor** (the nonzero x such that *ax* = 0 may be different from the nonzero y 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**. If there are no nontrivial zero divisors in *R*, then *R* is a domain.

## Examples

- In the ring , the residue class is a zero divisor since .
- The only zero divisor of the ring of integers is 0.
- A nilpotent element of a nonzero ring is always a two-sided zero divisor.
- An 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 other than 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 n-by-n matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of n-by-n 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 a 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
*R*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
*R*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 nonzero commutative ring
*R*, the set of non-zero-divisors is a multiplicative set in R. (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 Noetherian ring
*R*, the set of zero divisors is the union of the associated prime ideals of*R*.

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 R be a commutative ring, let M be an R-module, and let a be an element of R. One says that a is **M-regular** if the multiplication by a map is injective, and that a is a **zero divisor on M** otherwise.^{[4]} The set of M-regular elements is a multiplicative set in R.^{[5]}

Specializing the definitions of "M-regular" and "zero divisor on M" to the case *M* = *R* 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

- ↑ See Bourbaki, p. 98.
- ↑ Since the map is not injective, we have ax = ay, in which x differs from y, and thus a(x-y) = 0.
- ↑ See Lanski (2005).
- ↑ Matsumura, p. 12
- ↑ Matsumura, p. 12

## References

- N. Bourbaki (1989),
*Algebra I, Chapters 1–3*, Springer-Verlag. - Hazewinkel, Michiel, ed. (2001), "Zero divisor",
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - Michiel Hazewinkel; Nadiya Gubareni; Nadezhda Mikhaĭlovna Gubareni; Vladimir V. Kirichenko. (2004),
*Algebras, rings and modules*, Vol. 1, Springer, ISBN 1-4020-2690-0 - Charles Lanski (2005),
*Concepts in Abstract Algebra*, American Mathematical Soc., p. 342 - Hideyuki Matsumura (1980),
*Commutative algebra, 2nd edition*, The Benjamin/Cummings Publishing Company, Inc. - Weisstein, Eric W. "Zero Divisor".
*MathWorld*.