Zero divisor

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, or equivalently if the map from R to R that sends x to ax is not injective. 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. 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.

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.
• A idempotent element $e\ne 1$ of a ring is always a two-sided zero divisor, since $e(1-e)=0=(1-e)e$.
• 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 R₁ × R₂ with each R_i nonzero, (1,0)(0,1) = (0,0), so (1,0) is a zero divisor.

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 $x\ne0\ne y$, then $\begin{pmatrix}x&y\\0&z\end{pmatrix}$ is a left zero divisor iff $x$ 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 iff $z$ 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 $(a1,a2,a3,...)$. 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 $\mathrm{End}(S)$, the endomorphism ring 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 the projection map onto the first factor $P(a1,a2,a3,...)=(a1,0,0,...)$. 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. Note also that $RL$ is a two-sided zero-divisor since $RLP=0=PRL$, while $LR=1$ is not in any direction.

Non-examples

• The ring of integers modulo a prime number has no zero divisors other than 0. Since every nonzero element is a unit, this ring is a field.

• More generally, a division ring has no zero divisors except 0.

• A nonzero commutative ring whose only zero divisor is 0 is called an integral domain.

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 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.

See also

• Zero-product property
• Glossary of commutative algebra (Exact zero divisor)

References

• .
• Weisstein, Eric W., "Zero Divisor", MathWorld.