# Zero divisor

In abstract algebra, a non-zero element *a* of a ring *R* is a **left zero divisor** if there exists a non-zero *b* such that *ab* = 0. **Right zero divisors** are defined analogously. An element that is both a left and a right zero divisor is simply called a **zero divisor**. If the multiplication is commutative, then one does not have to distinguish between left and right zero divisors. A non-zero element that is neither left nor right zero divisor is called **regular**.

## Examples

The ring **Z** of integers does not have any zero divisors, but in the ring **Z**^{2} (where addition and multiplication are carried out component wise), we have (0,1) × (1,0) = (0,0) and so both (0,1) and (1,0) are zero divisors.

In the factor ring **Z**/6**Z**, the class of 4 is a zero divisor, since 3×4 is congruent to 0 modulo 6.

An example of a zero divisor in the ring of 2-by-2 matrices is the matrix

because for instance

## Properties

Left or right zero divisors can never be units, because if *a* is invertible and *ab* = 0, then 0 = *a*^{−1}0 = *a*^{−1}*ab* = *b*.

Every non-zero idempotent element *a*≠1 is a zero divisor, since *a*^{2} = *a* implies *a*(*a* − 1) = (*a* − 1)*a* = 0. Non-zero nilpotent ring elements are also trivially zero divisors.

In the ring of *n*-by-*n* matrices over some field, the left and right zero divisors coincide; they are precisely the nonzero singular matrices. In the ring of *n*-by-*n* matrices over some integral domain, the zero divisors are precisely the nonzero matrices with determinant zero.

If *a* is a left zero divisor, and *x* is an arbitrary ring element, then *xa* is either zero or a left zero divisor. The following example shows that the same cannot be said about *ax*. Consider the set of ∞-by-∞ matrices over the ring of integers, where every row and every column contains only finitely many non-zero entries. This is a ring with ordinary matrix multiplication. The matrix

is a left zero divisor and *B* = *A*^{T} is therefore a right zero divisor. But *AB* is the identity matrix and hence certainly not a zero divisor. In particular, we can conclude that *A* cannot be a right zero divisor.

A commutative ring with 0≠1 and without zero divisors is called an integral domain.

Zero divisors occur in **Z**/n**Z** if and only if n is composite. When n is prime, there are no zero divisors and this factor ring is, in fact, a field, as every element is a unit.

Zero divisors also occur in the sedenions, or 16- dimensional hypercomplex numbers under the Cayley-Dickson construction.

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