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Zero divisor

Summary

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.

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 ${\displaystyle e\neq 1}$  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 ${\displaystyle K[G]}$ .

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 ${\displaystyle x\neq 0\neq z}$ , 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 × 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.

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 = 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.

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.[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.