# Zero divisor

In abstract algebra, an element *a* of a ring *R* is called a **left zero divisor** if there exists a nonzero *x* in *R* such that *ax* = 0,^{[1]} or equivalently if the map from *R* to *R* that sends *x* to *ax* is not injective.^{[a]} Similarly, an element *a* of a ring is called a **right zero divisor** if there exists a nonzero *y* in *R* 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 *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 left zero divisor is called **left regular** or **left cancellable**. Similarly, an element of a ring that is not a right zero divisor is called **right regular** or **right cancellable**.
An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called **regular** or **cancellable**,^{[3]} or a **non-zero-divisor**. A zero divisor that is nonzero is called a **nonzero zero divisor** or a **nontrivial zero divisor**. A nonzero ring with no nontrivial zero divisors is called a domain.

## Examples[edit]

- In the ring , the residue class is a zero divisor since .
- The only zero divisor of the ring of integers is .
- 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 .
- The ring of matrices over a field has nonzero zero divisors if . 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 with each nonzero, , so is a zero divisor.
- Let be a field and be a group. Suppose that has an element of finite order . Then in the group ring one has , with neither factor being zero, so is a nonzero zero divisor in .

### One-sided zero-divisor[edit]

- Consider the ring of (formal) matrices with and . Then and . If , then is a left zero divisor if and only if is even, since , and it is a right zero divisor if and only if 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. is a two-sided zero-divisor since , while is not in any direction.

## Non-examples[edit]

- 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 finite 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[edit]

- 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 for some nonzero*x*, then 0 =*a*^{−1}0 =*a*^{−1}*ax*=*x*, a contradiction. - An element is cancellable on the side on which it is regular. That is, if
*a*is a left regular,*ax*=*ay*implies that*x*=*y*, and similarly for right regular.

## Zero as a zero divisor[edit]

There is no need for a separate convention for 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 any nonzero element x satisfies 0*x*= 0 =*x*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.

Some references include or exclude 0 as a zero divisor in *all* rings by convention, but they then suffer from having to introduce exceptions in statements such as the following:

- In a 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*.

## Zero divisor on a module[edit]

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.^{[4]}

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[edit]

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

## Notes[edit]

**^**Since the map is not injective, we have*ax*=*ay*, in which*x*differs from*y*, and thus*a*(*x*−*y*) = 0.

## References[edit]

**^**N. Bourbaki (1989),*Algebra I, Chapters 1–3*, Springer-Verlag, p. 98**^**Charles Lanski (2005),*Concepts in Abstract Algebra*, American Mathematical Soc., p. 342**^**Nicolas Bourbaki (1998).*Algebra I*. Springer Science+Business Media. p. 15.- ^
^{a}^{b}Hideyuki Matsumura (1980),*Commutative algebra, 2nd edition*, The Benjamin/Cummings Publishing Company, Inc., p. 12

## Further reading[edit]

- "Zero divisor",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Michiel Hazewinkel; Nadiya Gubareni; Nadezhda Mikhaĭlovna Gubareni; Vladimir V. Kirichenko. (2004),
*Algebras, rings and modules*, Vol. 1, Springer, ISBN 1-4020-2690-0`|volume=`

has extra text (help) - Weisstein, Eric W. "Zero Divisor".
*MathWorld*.