In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero divisor if there exists a nonzero in such that . 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. An element that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero such that may be different from the nonzero such that ). 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 (respectively, not a right zero divisor) is called left regular or left cancellable (respectively, 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, or a non-zero-divisor. (N.B.: In "non-zero-divisor", the prefix "non-" is understood to modify "zero-divisor" as a whole rather than just the word "zero". In some texts, "zero divisor" is written as "zerodivisor" and "non-zero-divisor" as "nonzerodivisor" or "non-zerodivisor" for clarity). A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. A non-zero ring with no nontrivial zero divisors is called a domain.

Examples

• In the ring \mathbb{Z}/4\mathbb{Z}.

• of integers.

• A nilpotent element of a nonzero ring is always a two-sided zero divisor.

• An idempotent element.

• The ring of n × n matrices over a field has nonzero zero divisors if n ≥ 2. Examples of zero divisors in the ring of 2 × 2 matrices (over any nonzero ring) are shown here:

• A direct product is a zero divisor.

• be a group of finite order. Then in the group ring.

One-sided zero-divisor

• is a left zero divisor if and only if is even, then it is a two-sided zero-divisor.

• be the set of all sequences. Take for the ring all additive maps, with pointwise addition and composition, the endomorphism ring.) Three examples of elements of this ring are the right shift, the left shift, and the projection map is not in any direction.

Non-examples

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

• More generally, a division ring has no nonzero zero divisors.

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

Properties

• In the ring of × matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of × matrices over an integral domain, the zero divisors are precisely the matrices with determinant zero.

• Left or right zero divisors can never be unitss, because if is invertible and for some nonzero , then , a contradiction.

• An element is cancellable on the side on which it is regular. That is, if is a left regular, implies that , and similarly for right regular.

Zero as a zero divisor

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

• If is a ring other than the zero ring, then is a (two-sided) zero divisor, because any nonzero element satisfies .

• If is the zero ring, in which , then is not a zero divisor, because there is no nonzero element that when multiplied by yields .

Some references include or exclude 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 , the set of non-zero-divisors is a multiplicative set in . (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 , the set of zero divisors is the union of the associated prime ideals of .

Zero divisor on a module

Let be a commutative ring, let be an -module, and let be an element of . One says that is -regular if the "multiplication by is injective, and that is a zero divisor on otherwise. The set of -regular elements is a multiplicative set in .

Specializing the definitions of "-regular" and "zero divisor on " to the case 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)

• Zero-divisor graph

• Sedenions, which have zero divisors

Notes

References

Further reading