Definition:Zero Divisor
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.