In
abstract algebra, an
elementa of a
ringR 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, the residue class is a zero divisor since .
The only zero divisor of the ring of
integers is .
A
nilpotent element of a nonzero ring is always a two-sided zero divisor.
An
idempotent element of a ring is always a two-sided zero divisor, since .
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:
A
direct product of two or more nonzero rings always has nonzero zero divisors. For example, in with each nonzero, , so is a zero divisor.
Let be a field and be a
group. Suppose that has an element of finite
order. Then in the
group ring one has , with neither factor being zero, so is a nonzero zero divisor in .
One-sided zero-divisor
Consider the ring of (formal) matrices with and . Then and . If , then is a left zero divisor
if and only if is
even, since , and it is a right zero divisor if and only if is even for similar reasons. If either of is , 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 be the
set of all
sequences of integers . Take for the ring all
additive maps from to , with
pointwise addition and
composition as the ring operations. (That is, our ring is , the endomorphism ring of the additive group .) Three examples of elements of this ring are the right shift, the left shift, and the projection map onto the first factor . All three of these additive maps are not zero, and the composites and are both zero, so is a left zero divisor and is a right zero divisor in the ring of additive maps from to . However, is not a right zero divisor and is not a left zero divisor: the composite is the identity. is a two-sided zero-divisor since , while 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−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.
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 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.
In
abstract algebra, an
elementa of a
ringR 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, the residue class is a zero divisor since .
The only zero divisor of the ring of
integers is .
A
nilpotent element of a nonzero ring is always a two-sided zero divisor.
An
idempotent element of a ring is always a two-sided zero divisor, since .
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:
A
direct product of two or more nonzero rings always has nonzero zero divisors. For example, in with each nonzero, , so is a zero divisor.
Let be a field and be a
group. Suppose that has an element of finite
order. Then in the
group ring one has , with neither factor being zero, so is a nonzero zero divisor in .
One-sided zero-divisor
Consider the ring of (formal) matrices with and . Then and . If , then is a left zero divisor
if and only if is
even, since , and it is a right zero divisor if and only if is even for similar reasons. If either of is , 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 be the
set of all
sequences of integers . Take for the ring all
additive maps from to , with
pointwise addition and
composition as the ring operations. (That is, our ring is , the endomorphism ring of the additive group .) Three examples of elements of this ring are the right shift, the left shift, and the projection map onto the first factor . All three of these additive maps are not zero, and the composites and are both zero, so is a left zero divisor and is a right zero divisor in the ring of additive maps from to . However, is not a right zero divisor and is not a left zero divisor: the composite is the identity. is a two-sided zero-divisor since , while 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−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.
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 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.