Zero divisor

Template:Short description Template:Use American English 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.Template:Efn 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 (respectively, not a right zero divisor) is called left regular or left cancellable (respectively, 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,Template:Refn or a non-zero-divisor. (N.B.: In "non-zero-divisor", the prefix "non-" is understood to modify "zero-divisor" as a whole rather than just the word "zero". In some texts, "zero divisor" is written as "zerodivisor" and "non-zero-divisor" as "nonzerodivisor" or "non-zerodivisor" for clarity.) A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. A non-zero ring with no nontrivial zero divisors is called a domain.

Examples

• In the ring $\mathbb{\mathbb{Z}}/4\mathbb{\mathbb{Z}}$, the residue class $\bar{2}$ is a zero divisor since $\bar{2}\times \bar{2}=\bar{4}=\bar{0}$.
• The only zero divisor of the ring $\mathbb{\mathbb{Z}}$ of integers is $0$.
• A nilpotent element of a nonzero ring is always a two-sided zero divisor.
• An idempotent element $e\ne 1$ of a ring is always a two-sided zero divisor, since $e(1-e)=0=(1-e)e$.
• The ring of n × n matrices over a field has nonzero zero divisors if n ≥ 2. Examples of zero divisors in the ring of 2 × 2 matrices (over any nonzero ring) are shown here:

$$\left(\begin{array}{cc}1& 1\\ 2& 2\end{array}\right)\left(\begin{array}{cc}1& 1\\ -1& -1\end{array}\right)=\left(\begin{array}{cc}-2& 1\\ -2& 1\end{array}\right)\left(\begin{array}{cc}1& 1\\ 2& 2\end{array}\right)=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right),$$ $$\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)=\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right).$$

• 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 $\left(\begin{array}{cc}x& y\\ 0& z\end{array}\right)$ with $x,z\in \mathbb{\mathbb{Z}}$ and $y\in \mathbb{\mathbb{Z}}/2\mathbb{\mathbb{Z}}$. Then $\left(\begin{array}{cc}x& y\\ 0& z\end{array}\right)\left(\begin{array}{cc}a& b\\ 0& c\end{array}\right)=\left(\begin{array}{cc}xa& xb+yc\\ 0& zc\end{array}\right)$ and $\left(\begin{array}{cc}a& b\\ 0& c\end{array}\right)\left(\begin{array}{cc}x& y\\ 0& z\end{array}\right)=\left(\begin{array}{cc}xa& ya+zb\\ 0& zc\end{array}\right)$. If $x\ne 0\ne z$, then $\left(\begin{array}{cc}x& y\\ 0& z\end{array}\right)$ is a left zero divisor if and only if $x$ is even, since $\left(\begin{array}{cc}x& y\\ 0& z\end{array}\right)\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)=\left(\begin{array}{cc}0& x\\ 0& 0\end{array}\right)$, 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 $\mathrm{E}\mathrm{n}\mathrm{d}(S)$, 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.

Non-examples

• The ring of integers modulo a prime number has no nonzero zero divisors. Since every nonzero element is a unit, this ring is a finite field.
• More generally, a division ring has no nonzero zero divisors.
• A non-zero commutative ring whose only zero divisor is 0 is called an integral domain.

Properties

• In the ring of Template:Mvar × Template:Mvar matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of Template:Mvar × Template:Mvar 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 Template:Mvar satisfies 0x = 0 = x 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.

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 Template:Mvar. (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 Template:Mvar be a commutative ring, let Template:Mvar be an Template:Mvar-module, and let Template:Mvar be an element of Template:Mvar. One says that Template:Mvar is Template:Mvar-regular if the "multiplication by Template:Mvar" map $M\stackrel{a}{\to }M$ is injective, and that Template:Mvar is a zero divisor on Template:Mvar otherwise. The set of Template:Mvar-regular elements is a multiplicative set in Template:Mvar.

Specializing the definitions of "Template:Mvar-regular" and "zero divisor on Template:Mvar" 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)
• Zero-divisor graph
• Sedenions, which have zero divisors

Further reading

• Template:Springer