Definition:Zero Divisor

From ProofWiki
Jump to: navigation, search

Definition

Rings

Let $\left({R, +, \circ}\right)$ 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 $R \setminus \left\{{0_R}\right\}$.


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:

$x \mathrel \backslash 0_R$


Commutative Rings

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


Let $\left({R, +, \circ}\right)$ 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 $R \setminus \left\{{0_R}\right\}$.


The expression:

$x$ is a zero divisor

can be written:

$x \mathrel \backslash 0_R$


Algebras

Let $\left({A_R, \oplus}\right)$ be an algebra over a ring $\left({R, +, \cdot}\right)$.

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


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

Then $a$ and $b$ are zero divisors of $A_R$ iff $a \oplus b = \mathbf 0_R$.