Category:Definitions/Associates
This category contains definitions related to Associates in the context of Abstract Algebra.
Related results can be found in Category:Associates.
Let $\struct {D, +, \circ}$ be an integral domain.
Let $x, y \in D$.
Definition 1
$x$ is an associate of $y$ (in $D$) if and only if they are both divisors of each other.
That is, $x$ and $y$ are associates (in $D$) if and only if $x \divides y$ and $y \divides x$.
Definition 2
$x$ and $y$ are associates (in $D$) if and only if:
- $\ideal x = \ideal y$
where $\ideal x$ and $\ideal y$ denote the ideals generated by $x$ and $y$ respectively.
Definition 3
$x$ and $y$ are associates (in $D$) if and only if there exists a unit $u$ of $\struct {D, +, \circ}$ such that:
- $y = u \circ x$
and consequently:
- $x = u^{-1} \circ y$
That is, if and only if $x$ and $y$ are unit multiples of each other.
Pages in category "Definitions/Associates"
The following 9 pages are in this category, out of 9 total.
A
- Definition:Associate
- Definition:Associate in Integral Domain
- Definition:Associate of Integer
- Definition:Associate/Commutative and Unitary Ring
- Definition:Associate/Integers
- Definition:Associate/Integral Domain
- Definition:Associate/Integral Domain/Definition 1
- Definition:Associate/Integral Domain/Definition 2
- Definition:Associate/Integral Domain/Definition 3