Zero divisor explained

Zero divisor should not be confused with Division by zero.

In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that, or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero divisor if there exists a nonzero in such that . 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  that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero such that may be different from the nonzero such that). 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, 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

\overline{2}

is a zero divisor since

\overline{2} x \overline{2}=\overline{4}=\overline{0}

.

Z

of integers is

0

.

e\ne1

of a ring is always a two-sided zero divisor, since

e(1-e)=0=(1-e)e

.

\begin1&1\\2&2\end\begin1&1\\-1&-1\end=\begin-2&1\\-2&1\end\begin1&1\\2&2\end=\begin0&0\\0&0\end, \begin1&0\\0&0\end\begin0&0\\0&1\end=\begin0&0\\0&1\end\begin1&0\\0&0\end=\begin0&0\\0&0\end.

R1 x R2

with each

Ri

nonzero,

(1,0)(0,1)=(0,0)

, so

(1,0)

is a zero divisor.

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++gn-1)=1-gn=0

, with neither factor being zero, so

1-g

is a nonzero zero divisor in

K[G]

.

One-sided zero-divisor

\begin{pmatrix}x&y\\0&z\end{pmatrix}

with

x,z\inZ

and

y\inZ/2Z

. 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\ne0\nez

, 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.

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

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.

RL

is a two-sided zero-divisor since

RLP=0=PRL

, while

LR=1

is not in any direction.

Non-examples

Properties

Zero as a zero divisor

There is no need for a separate convention for the case, because the definition applies also in this case:

Some references include or exclude as a zero divisor in all rings by convention, but they then suffer from having to introduce exceptions in statements such as the following:

Zero divisor on a module

Let be a commutative ring, let be an -module, and let be an element of . One says that is -regular if the "multiplication by " map

M\stackrel{a}\toM

is injective, and that is a zero divisor on otherwise. The set of -regular elements is a multiplicative set in .

Specializing the definitions of "-regular" and "zero divisor on " to the case recovers the definitions of "regular" and "zero divisor" given earlier in this article.

See also