# 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, 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. 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, 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

• In the ring $\mathbb {Z} /4\mathbb {Z}$ , the residue class ${\overline {2}}$ is a zero divisor since ${\overline {2}}\times {\overline {2}}={\overline {4}}={\overline {0}}$ .
• The only zero divisor of the ring $\mathbb {Z}$ of integers is $0$ .
• A nilpotent element of a nonzero ring is always a two-sided zero divisor.
• An idempotent element $e\neq 1$ of a ring is always a two-sided zero divisor, since $e(1-e)=0=(1-e)e$ .
• The ring of $n\times n$ matrices over a field has nonzero zero divisors if $n\geq 2$ . Examples of zero divisors in the ring of $2\times 2$ matrices (over any nonzero ring) are shown here:
${\begin{pmatrix}1&1\\2&2\end{pmatrix}}{\begin{pmatrix}1&1\\-1&-1\end{pmatrix}}={\begin{pmatrix}-2&1\\-2&1\end{pmatrix}}{\begin{pmatrix}1&1\\2&2\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}},$ ${\begin{pmatrix}1&0\\0&0\end{pmatrix}}{\begin{pmatrix}0&0\\0&1\end{pmatrix}}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}{\begin{pmatrix}1&0\\0&0\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}$ .
• A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in $R_{1}\times R_{2}$ with each $R_{i}$ nonzero, $(1,0)(0,1)=(0,0)$ , so $(1,0)$ is a zero divisor.
• Let $K$ be a field and $G$ be a group. Suppose that $G$ has an element $g$ of finite order $n>1$ . Then in the group ring $K[G]$ one has $(1-g)(1+g+\cdots +g^{n-1})=1-g^{n}=0$ , with neither factor being zero, so $1-g$ is a nonzero zero divisor in $K[G]$ .

### One-sided zero-divisor

• Consider the ring of (formal) matrices ${\begin{pmatrix}x&y\\0&z\end{pmatrix}}$ with $x,z\in \mathbb {Z}$ and $y\in \mathbb {Z} /2\mathbb {Z}$ . Then ${\begin{pmatrix}x&y\\0&z\end{pmatrix}}{\begin{pmatrix}a&b\\0&c\end{pmatrix}}={\begin{pmatrix}xa&xb+yc\\0&zc\end{pmatrix}}$ and ${\begin{pmatrix}a&b\\0&c\end{pmatrix}}{\begin{pmatrix}x&y\\0&z\end{pmatrix}}={\begin{pmatrix}xa&ya+zb\\0&zc\end{pmatrix}}$ . If $x\neq 0\neq z$ , then ${\begin{pmatrix}x&y\\0&z\end{pmatrix}}$ is a left zero divisor if and only if $x$ is even, since ${\begin{pmatrix}x&y\\0&z\end{pmatrix}}{\begin{pmatrix}0&1\\0&0\end{pmatrix}}={\begin{pmatrix}0&x\\0&0\end{pmatrix}}$ , and it is a right zero divisor if and only if $z$ is even for similar reasons. If either of $x,z$ is $0$ , 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 $S$ be the set of all sequences of integers $(a_{1},a_{2},a_{3},...)$ . 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 $\mathrm {End} (S)$ , the endomorphism ring of the additive group $S$ .) Three examples of elements of this ring are the right shift $R(a_{1},a_{2},a_{3},...)=(0,a_{1},a_{2},...)$ , the left shift $L(a_{1},a_{2},a_{3},...)=(a_{2},a_{3},a_{4},...)$ , and the projection map onto the first factor $P(a_{1},a_{2},a_{3},...)=(a_{1},0,0,...)$ . All three of these additive maps are not zero, and the composites $LP$ and $PR$ 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 $LR$ is the identity. $RL$ is a two-sided zero-divisor since $RLP=0=PRL$ , while $LR=1$ is not in any direction.

## Properties

• 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, then 0 = a−10 = a−1ax = x for some nonzero x.
• 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

There is no need for a separate convention regarding 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 0 · a = 0 = a · 0, where a is a nonzero element of R.
• 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.

Such properties are needed in order to make the following general statements true:

• 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.

Some references choose to exclude 0 as a zero divisor by convention, but then they must introduce exceptions in the two general statements just made.

## Zero divisor on a module

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 $M{\stackrel {a}{\to }}M$ is injective, and that a is a zero divisor on M otherwise. The set of M-regular elements is a multiplicative set in R.

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.