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 (one to one).[lower-alpha 1] 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.[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 is called left regular or left cancellable. Similarly, an element of a ring that is not a right zero divisor is called 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. 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 $e(1-e)=0=(1-e)e$ .
The ring of $n\times n$ matrices over a field has nonzero zero divisors if $n\geq 2$ . 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 $R_{1}\times R_{2}$ 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 $x\neq 0\neq z$ , then ${\begin{pmatrix}x&y\\0&z\end{pmatrix}}$ is a left zero divisor if and only if $x$ 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 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 $\mathrm {End} (S)$ , 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 zero divisors other than 0. Since every nonzero element is a unit, this ring is a finite 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 $0 = a -1 0 = a -1 ax = x$ for some nonzero $x$ .
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 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 $0 \cdot a = 0 = a \cdot 0$ , where $a$ is a nonzero element of $R$ .
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.[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.

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Notes

Since the map is not injective, we have $ax = ay$ , in which $x$ differs from $y$ , and thus $a (x - y) = 0$ .

References

N. Bourbaki (1989), Algebra I, Chapters 1–3, Springer-Verlag, p. 98
Charles Lanski (2005), Concepts in Abstract Algebra, American Mathematical Soc., p. 342
Nicolas Bourbaki (1998). Algebra I. Springer Science+Business Media. p. 15.
Hideyuki Matsumura (1980), Commutative algebra, 2nd edition, The Benjamin/Cummings Publishing Company, Inc., p. 12