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.^{[lower-alpha 1]} 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 $ℤ / 4 ℤ$ , 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 $ℤ$ 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 × 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:

$(\begin{matrix} 1 & 1 \\ 2 & 2 \end{matrix}) (\begin{matrix} 1 & 1 \\ - 1 & - 1 \end{matrix}) = (\begin{matrix} - 2 & 1 \\ - 2 & 1 \end{matrix}) (\begin{matrix} 1 & 1 \\ 2 & 2 \end{matrix}) = (\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}),$ $(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}) (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}) = (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}) (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}) = (\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}) .$

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 $(1 - g) (1 + g + \dots + g^{n - 1}) = 1 - g^{n} = 0$ , 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 $(\begin{matrix} x & y \\ 0 & z \end{matrix})$ with $x, z \in ℤ$ and $y \in ℤ / 2 ℤ$ . Then $(\begin{matrix} x & y \\ 0 & z \end{matrix}) (\begin{matrix} a & b \\ 0 & c \end{matrix}) = (\begin{matrix} x a & x b + y c \\ 0 & z c \end{matrix})$ and $(\begin{matrix} a & b \\ 0 & c \end{matrix}) (\begin{matrix} x & y \\ 0 & z \end{matrix}) = (\begin{matrix} x a & y a + z b \\ 0 & z c \end{matrix})$ . If $x \neq 0 \neq z$ , then $(\begin{matrix} x & y \\ 0 & z \end{matrix})$ is a left zero divisor if and only if $x$ is even, since $(\begin{matrix} x & y \\ 0 & z \end{matrix}) (\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}) = (\begin{matrix} 0 & x \\ 0 & 0 \end{matrix})$ , 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 $E n d (S)$ , 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 $L P$ and $P R$ 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 $L R$ is the identity. $R L$ is a two-sided zero-divisor since $R L P = 0 = P R L$ , while $L R = 1$ 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 $n$ × $n$ 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 $0 = a -1 0 = a -1 ax = x$ , 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 $0 x = 0 = x 0$ .
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 $M \overset{a}{\to} M$ 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.

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.
↑ ^4.0 ^4.1 Hideyuki Matsumura (1980), Commutative algebra, 2nd edition, The Benjamin/Cummings Publishing Company, Inc., p. 12

Zero divisor

Contents

Examples

One-sided zero-divisor

Non-examples

Properties

Zero as a zero divisor

Zero divisor on a module

See also

Notes

References

Further reading

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