# Zero divisor

(Redirected from Zero divisors)

In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x such that ax = 0,[1] or equivalently if the map from R to R that sends x to ax is not injective.[2] Similarly, an element a of a ring is called a right zero divisor if there exists a nonzero y 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.[3] 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 zero divisor is called regular, 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 R, then R is 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 0.
• A nilpotent element of a nonzero ring is always a two-sided zero divisor.
• An idempotent element ${\displaystyle e\neq 1}$ of a ring is always a two-sided zero divisor, since ${\displaystyle e(1-e)=0=(1-e)e}$.
• 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 R1 × R2 with each Ri nonzero, (1,0)(0,1) = (0,0), so (1,0) is a zero divisor.

### 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 ${\displaystyle x\neq 0\neq y}$, then ${\displaystyle {\begin{pmatrix}x&y\\0&z\end{pmatrix}}}$ is a left zero divisor iff ${\displaystyle x}$ 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 iff ${\displaystyle z}$ 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 ${\displaystyle (a1,a2,a3,...)}$. 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 ${\displaystyle R(a1,a2,a3,...)=(0,a1,a2,...)}$, the left shift ${\displaystyle L(a1,a2,a3,...)=(a2,a3,a4,...)}$, and the projection map onto the first factor ${\displaystyle P(a1,a2,a3,...)=(a1,0,0,...)}$. 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. Note also that ${\displaystyle RL}$ 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, then 0 = a−10 = a−1ax = x, whereas x must be nonzero.

## 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 · 1 = 0 and 1 · 0 = 0.
• 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 nonzero 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 ${\displaystyle M{\stackrel {a}{\to }}M}$ is injective, and that a is a zero divisor on M otherwise.[4] The set of M-regular elements is a multiplicative set in R.[5]

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.