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,^[1] 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.^[2] 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,^[3] or a non-zero-divisor. 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.

• 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 ${\displaystyle e\neq 1}$ of a ring is always a two-sided zero divisor, since ${\displaystyle e(1-e)=0=(1-e)e}$.
• 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:

$${\displaystyle {\begin{pmatrix}1&1\\2&2\end{pmatrix}}{\begin{pmatrix}1&1\\-1&-1\end{pmatrix}}={\begin{pmatrix}-2&1\\-2&1\end{pmatrix}}{\begin{pmatrix}1&1\\2&2\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}},}$$ $${\displaystyle {\begin{pmatrix}1&0\\0&0\end{pmatrix}}{\begin{pmatrix}0&0\\0&1\end{pmatrix}}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}{\begin{pmatrix}1&0\\0&0\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}.}$$

• 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 ${\displaystyle n>1}$. 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 ${\displaystyle K[G]}$.

• 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 ${\displaystyle x\neq 0\neq z}$, 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.

• 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.

• In the ring of n × n matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of n × 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.

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 = 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 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.

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.^[4]

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.

• Zero-product property
• Glossary of commutative algebra (Exact zero divisor)
• Zero-divisor graph
• Sedenions, which have zero divisors

• "Zero divisor", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Michiel Hazewinkel; Nadiya Gubareni; Nadezhda Mikhaĭlovna Gubareni; Vladimir V. Kirichenko. (2004), Algebras, rings and modules, vol. 1, Springer, ISBN 1-4020-2690-0
• Weisstein, Eric W. "Zero Divisor". MathWorld.