In abstract algebra, an element ofa ring is called a left zero divisor if there exists a nonzerosuch 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 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. 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, 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, then is a domain.
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 are shown here:
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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.
One-sided zero-divisor
Consider the ring of 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. 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.
In the ring of -by- matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of -by- 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 is invertible and, then for some nonzero.
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 regarding the case, because the definition applies also in this case:
If is a ring other than the zero ring, then is a zero divisor, because, where is a nonzero element of.
If is the zero ring, in which, then is not a zero divisor, because there is no nonzero element that when multiplied by yields.
Such properties are needed in order to make the following general statements true:
In a commutative ring, the set of non-zero-divisors is a multiplicative set in. 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.
Let be a commutative ring, let be an -module, and let be an element of. One says that is -regular if the "multiplication by " map 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.
