# Zero divisor

Short description: Ring element that can be multiplied by a non-zero element to equal 0

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.[lower-alpha 1] 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 $\displaystyle{ \mathbb{Z}/4\mathbb{Z} }$, the residue class $\displaystyle{ \overline{2} }$ is a zero divisor since $\displaystyle{ \overline{2} \times \overline{2}=\overline{4}=\overline{0} }$.
• The only zero divisor of the ring $\displaystyle{ \mathbb{Z} }$ of integers is $\displaystyle{ 0 }$.
• A nilpotent element of a nonzero ring is always a two-sided zero divisor.
• An idempotent element $\displaystyle{ e\ne 1 }$ of a ring is always a two-sided zero divisor, since $\displaystyle{ e(1-e)=0=(1-e)e }$.
• The ring of $\displaystyle{ n \times n }$ matrices over a field has nonzero zero divisors if $\displaystyle{ n \geq 2 }$. Examples of zero divisors in the ring of $\displaystyle{ 2\times 2 }$ matrices (over any nonzero ring) are shown here: $\displaystyle{ \begin{pmatrix}1&1\\2&2\end{pmatrix}\begin{pmatrix}1&1\\-1&-1\end{pmatrix}=\begin{pmatrix}-2&1\\-2&1\end{pmatrix}\begin{pmatrix}1&1\\2&2\end{pmatrix}=\begin{pmatrix}0&0\\0&0\end{pmatrix} , }$ $\displaystyle{ \begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&0\\0&1\end{pmatrix} =\begin{pmatrix}0&0\\0&1\end{pmatrix}\begin{pmatrix}1&0\\0&0\end{pmatrix} =\begin{pmatrix}0&0\\0&0\end{pmatrix}. }$
• A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in $\displaystyle{ R_1 \times R_2 }$ with each $\displaystyle{ R_i }$ nonzero, $\displaystyle{ (1,0)(0,1) = (0,0) }$, so $\displaystyle{ (1,0) }$ is a zero divisor.
• Let $\displaystyle{ K }$ be a field and $\displaystyle{ G }$ be a group. Suppose that $\displaystyle{ G }$ has an element $\displaystyle{ g }$ of finite order $\displaystyle{ n\gt 1 }$. Then in the group ring $\displaystyle{ K[G] }$ one has $\displaystyle{ (1-g)(1+g+ \cdots +g^{n-1})=1-g^{n}=0 }$, with neither factor being zero, so $\displaystyle{ 1-g }$ is a nonzero zero divisor in $\displaystyle{ K[G] }$.

### One-sided zero-divisor

• Consider the ring of (formal) matrices $\displaystyle{ \begin{pmatrix}x&y\\0&z\end{pmatrix} }$ with $\displaystyle{ x,z\in\mathbb{Z} }$ and $\displaystyle{ y\in\mathbb{Z}/2\mathbb{Z} }$. Then $\displaystyle{ \begin{pmatrix}x&y\\0&z\end{pmatrix}\begin{pmatrix}a&b\\0&c\end{pmatrix}=\begin{pmatrix}xa&xb+yc\\0&zc\end{pmatrix} }$ and $\displaystyle{ \begin{pmatrix}a&b\\0&c\end{pmatrix}\begin{pmatrix}x&y\\0&z\end{pmatrix}=\begin{pmatrix}xa&ya+zb\\0&zc\end{pmatrix} }$. If $\displaystyle{ x\ne0\ne z }$, then $\displaystyle{ \begin{pmatrix}x&y\\0&z\end{pmatrix} }$ is a left zero divisor if and only if $\displaystyle{ x }$ is even, since $\displaystyle{ \begin{pmatrix}x&y\\0&z\end{pmatrix}\begin{pmatrix}0&1\\0&0\end{pmatrix}=\begin{pmatrix}0&x\\0&0\end{pmatrix} }$, and it is a right zero divisor if and only if $\displaystyle{ z }$ is even for similar reasons. If either of $\displaystyle{ x,z }$ is $\displaystyle{ 0 }$, 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 $\displaystyle{ S }$ be the set of all sequences of integers $\displaystyle{ (a_1,a_2,a_3,...) }$. Take for the ring all additive maps from $\displaystyle{ S }$ to $\displaystyle{ S }$, with pointwise addition and composition as the ring operations. (That is, our ring is $\displaystyle{ \mathrm{End}(S) }$, the endomorphism ring of the additive group $\displaystyle{ S }$.) Three examples of elements of this ring are the right shift $\displaystyle{ R(a_1,a_2,a_3,...)=(0,a_1,a_2,...) }$, the left shift $\displaystyle{ L(a_1,a_2,a_3,...)=(a_2,a_3,a_4,...) }$, and the projection map onto the first factor $\displaystyle{ P(a_1,a_2,a_3,...)=(a_1,0,0,...) }$. All three of these additive maps are not zero, and the composites $\displaystyle{ LP }$ and $\displaystyle{ PR }$ are both zero, so $\displaystyle{ L }$ is a left zero divisor and $\displaystyle{ R }$ is a right zero divisor in the ring of additive maps from $\displaystyle{ S }$ to $\displaystyle{ S }$. However, $\displaystyle{ L }$ is not a right zero divisor and $\displaystyle{ R }$ is not a left zero divisor: the composite $\displaystyle{ LR }$ is the identity. $\displaystyle{ RL }$ is a two-sided zero-divisor since $\displaystyle{ RLP=0=PRL }$, while $\displaystyle{ LR=1 }$ 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 for some nonzero x, then 0 = a−10 = a−1ax = x, a contradiction.
• 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 for 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 any nonzero element x satisfies 0x = 0 = x0.
• 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.

Some references include or exclude 0 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 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.

## 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 $\displaystyle{ M \,\stackrel{a}\to\, M }$ 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.