Category:Definitions/Zero Divisors
This category contains definitions related to Zero Divisors.
Related results can be found in Category: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:
- $a \oplus b = \mathbf 0_R$
Pages in category "Definitions/Zero Divisors"
The following 19 pages are in this category, out of 19 total.
P
Z
- Definition:Zero Divisor
- Definition:Zero Divisor of Algebra
- Definition:Zero Divisor of Commutative Ring
- Definition:Zero Divisor of Ring
- Definition:Zero Divisor/Algebra
- Definition:Zero Divisor/Also defined as
- Definition:Zero Divisor/Also known as
- Definition:Zero Divisor/Commutative Ring
- Definition:Zero Divisor/Ring
- Definition:Zero Divisor/Warning
- Definition:Zero-Divisor
- Definition:Zerodivisor