# Zero divisor

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In abstract algebra, two nonzero elements a and b of a ring are respectively called a left zero divisor and a right zero divisor if a b = 0;[1] 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 w that is both a left and a right zero divisor,[3] is called a two-sided zero divisor. If the ring is commutative, then the left and right zero divisors are the same. A non-zero element of a ring that is not a zero divisor is called regular.

## Examples

• The ring Z of integers has no zero divisors, but in the ring Z/4Z the number 2 is a zero divisor: 2 × 2 = 4 = 0 as a divisor of 4, which is a composite number.
• A nonzero nilpotent element is always a zero-divisor (left and right).
• A nonzero idempotent element is always a two-sided zero divisor, provided that it is not 1.
• An example of a zero divisor in the ring of 2-by-2 matrices (over any unital ring except trivial) is the matrix
$\begin{pmatrix}1&1\\ 2&2\end{pmatrix}$
because for instance
$\begin{pmatrix}1&1\\ 2&2\end{pmatrix}\begin{pmatrix}1&1\\ -1&-1\end{pmatrix}=\begin{pmatrix}-2&1\\ -2&1\end{pmatrix}\begin{pmatrix}1&1\\ 2&2\end{pmatrix}=\begin{pmatrix}0&0\\ 0&0\end{pmatrix}.$
• Actually, the simplest example of a pair of zero divisor matrices is
$\begin{pmatrix}1&0\\0&0\end{pmatrix} \begin{pmatrix}0&0\\0&1\end{pmatrix} = \begin{pmatrix}0&0\\0&0\end{pmatrix} = \begin{pmatrix}0&0\\0&1\end{pmatrix} \begin{pmatrix}1&0\\0&0\end{pmatrix}.$
• A direct product of two or more non-trivial rings always has zero divisors similarly to the 2 × 2-matrix example just above (the ring of diagonal 2 × 2 matrices over a ring R is the same as the direct product R × R).
• Here is an example of a ring with an element that is a zero divisor on one side only. Let S be the set of all sequences of integers (a1, a2, a3, …). Take for the ring all additive maps from S to S, with pointwise addition and composition as the ring operations. (That is, our ring is End(S), the endomorphisms of the additive group S.) Three examples of elements of this ring are the right shift R(a1, a2, a3, …) = (0, a1, a2, …), the left shift L(a1, a2, a3, …) = (a2, a3, a4, …), and a third additive map T(a1, a2, a3, …) = (a1, 0, 0, …). All three of these additive maps are not zero, and the composites L T and T R are both zero, so L is a left zero divisor and R is a right zero divisor in the ring of additive maps from S to S. However, L is not a right zero divisor and R is not a left zero divisor: the composite L R is the identity, so if some additive map f from S to S satisfies f L= 0 then composing both sides of this equation on the right with R shows (f L) R = f (L R) = f 1 = f has to be 0, and similarly if some f satisfies R f = 0 then composing both sides on the left with L shows f is 0.

Continuing with this example, note that while R L is a left zero divisor ((R L) T = R (L T) is 0 because L T is), L R is not a zero divisor on either side because it is the identity.

Concretely, we can interpret additive maps from S to S as countably infinite matrices. The matrix

$A = \begin{pmatrix} 0 & 1 & 0 &0&0&\\ 0 & 0 & 1 &0&0&\cdots\\ 0 & 0 & 0 &1&0&\\ 0&0&0&0&1&\\ &&\vdots&&&\ddots \end{pmatrix}$

realizes L explicitly (just apply the matrix to a vector and see the effect is exactly a left shift) and the transpose B = AT realizes the right shift on S. That A B is the identity matrix is the same as saying L R is the identity. In particular, as matrices A is a left zero divisor but not a right zero divisor.

## Non-examples

The ring of integers modulo a prime number does not have zero divisors and this ring is, in fact, a field, as every non-zero element is a unit.

More generally, there are no zero divisors in division rings.

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

## Properties

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

Left or right zero divisors can never be units, because if a is invertible and a b = 0, then 0 = a−10 = a−1a b = b.

Every non-trivial idempotent element  a in a ring is a zero divisor, since a2 = a implies that a (a − 1) = (a − 1) a = 0, with nontriviality ensuring that neither factor is 0. Nonzero nilpotent ring elements are also trivially zero divisors.

The set of zero divisors is the union of the associated prime ideals of the ring.

## Notes

1. ^ See Hazewinkel et. al. (2004), p. 2.
2. ^ See Lanski (2005).
3. ^ "w is both a left and a right divisor" means v: vw = 0 and u: wu = 0, but such v and u are not necessarily equal.

## References

• 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