Category:Zero Divisors

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This category contains results about Zero Divisors.
Definitions specific to this category can be found in Definitions/Zero Divisors.

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:

$a \oplus b = \mathbf 0_R$