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, or equivalently if the map from R to R that sends x to ax is not injective. 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. 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.
In the ring
Z
/
4
Z
, the residue class
2
¯
is a zero divisor since
2
¯
×
2
¯
=
4
¯
=
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
≠
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
×
2
matrices (over any nonzero ring) are shown here:
(
1
1
2
2
)
(
1
1
−
1
−
1
)
=
(
−
2
1
−
2
1
)
(
1
1
2
2
)
=
(
0
0
0
0
)
,
(
1
0
0
0
)
(
0
0
0
1
)
=
(
0
0
0
1
)
(
1
0
0
0
)
=
(
0
0
0
0
)
.
A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in R1 × R2 with each Ri nonzero, (1,0)(0,1) = (0,0), so (1,0) is a zero divisor.
Consider the ring of (formal) matrices
(
x
y
0
z
)
with
x
,
z
∈
Z
and
y
∈
Z
/
2
Z
. Then
(
x
y
0
z
)
(
a
b
0
c
)
=
(
x
a
x
b
+
y
c
0
z
c
)
and
(
a
b
0
c
)
(
x
y
0
z
)
=
(
x
a
y
a
+
z
b
0
z
c
)
. If
x
≠
0
≠
y
, then
(
x
y
0
z
)
is a left zero divisor iff
x
is even, since
(
x
y
0
z
)
(
0
1
0
0
)
=
(
0
x
0
0
)
; 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
(
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. Note also that
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.
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.
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−10 = a−1ax = x, whereas x must be nonzero.
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 · 1 = 0 and 1 · 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.
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
→
a
M
is injective, and that a is a zero divisor on M otherwise. The set of M-regular elements is a multiplicative set in R.
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.
