Category:Associates
This category contains results about Associates in the context of Abstract Algebra.
Definitions specific to this category can be found in Definitions/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.
Subcategories
This category has only the following subcategory.
Pages in category "Associates"
The following 4 pages are in this category, out of 4 total.