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 $ax = 0$ ,^[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 $\mathbb {Z} /4\mathbb {Z}$ , 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.
An idempotent element $e\neq 1$ 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 $x\neq 0\neq y$ , 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 $0 = a -1 0 = a -1 ax = x$ , 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 $0 \cdot 1 = 0$ and $1 \cdot 0 = 0$ .
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 $M{\stackrel {a}{\to }}M$ 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.

Notes

↑ See Bourbaki, p. 98.
↑ Since the map is not injective, we have $ax$ = $ay$ , in which $x$ differs from $y$ , and thus $a$ ( $x$ - $y$ ) = 0.
↑ See Lanski (2005).
↑ Matsumura, p. 12
↑ Matsumura, p. 12

References

N. Bourbaki (1989), Algebra I, Chapters 1–3, Springer-Verlag .
Hazewinkel, Michiel, ed. (2001), "Zero divisor", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Michiel Hazewinkel; Nadiya Gubareni; Nadezhda Mikhaĭlovna Gubareni; Vladimir V. Kirichenko. (2004), Algebras, rings and modules, Vol. 1, Springer, ISBN 1-4020-2690-0
Charles Lanski (2005), Concepts in Abstract Algebra, American Mathematical Soc., p. 342
Hideyuki Matsumura (1980), Commutative algebra, 2nd edition, The Benjamin/Cummings Publishing Company, Inc.
Weisstein, Eric W. "Zero Divisor". MathWorld.

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