Definition:Zero Divisor

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Definition

Rings

Let $\struct {R, +, \circ}$ be a ring.


A zero divisor (in $R$) is an element $x \in R$ such that either:

$\exists y \in R^*: x \circ y = 0_R$

or:

$\exists y \in R^*: y \circ x = 0_R$

where $R^*$ is defined as $R \setminus \set {0_R}$.


That is, such that $x$ is either a left zero divisor or a right zero divisor.


The expression:

$x$ is a zero divisor

can be written:

$x \divides 0_R$


Commutative Rings

The definition is usually made when the ring in question is commutative:


Let $\struct {R, +, \circ}$ be a commutative ring.


A zero divisor (in $R$) is an element $x \in R$ such that:

$\exists y \in R^*: x \circ y = 0_R$

where $R^*$ is defined as $R \setminus \set {0_R}$.


The expression:

$x$ is a zero divisor

can be written:

$x \divides 0_R$


Algebras

Let $\struct {A_R, \oplus}$ be an algebra over a ring $\struct {R, +, \cdot}$.

Let the zero vector of $A_R$ be $\mathbf 0_R$.


Let $a, b \in A_R$ such that $b \ne \mathbf 0_R$.

Then $a$ is a zero divisor of $A_R$ if and only if:

$a \oplus b = \mathbf 0_R$


Also defined as

Some sources define a zero divisor as an element $x \in R_{\ne 0_R}$ such that:

$\exists y \in R_{\ne 0_R}: x \circ y = 0_R$

where $R_{\ne 0_R}$ is defined as $R \setminus \set {0_R}$.

That is, the element $0_R$ itself is not classified as a zero divisor.

This definition is the same as the one given on this website as a proper zero divisor.


Also known as

Some sources hyphenate, as: zero-divisor.

Some sources run the words together: zerodivisor.

Some use the more explicit and pedantic divisor of zero.


Warning

Beware the terminology divisor of zero.

It is easy to confuse this with the fact that, for every element $a$ of a ring $R$, from Ring Product with Zero we have that $a \circ 0_R = 0_R$.

Hence one may say that every such element is a divisor of zero.

However, the concept of a zero divisor specifically requires that the $b$ in $a \circ b = 0_R$ is not zero.


Also see

  • Results about zero divisors can be found here.