Definition:Associate
Definition
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.
Integers
As the integers form an integral domain, the definition can be applied directly to the set of integers $\Z$:
Let $x, y \in \Z$.
Then $x$ is an associate of $y$ if and only if they are both divisors of each other.
That is, $x$ and $y$ are associates if and only if $x \divides y$ and $y \divides x$.
Commutative and Unitary Ring
The concept of associatehood can also be applied to the general commutative and unitary ring, even though there may be (proper) zero divisors in the latter:
Let $\struct {R, +, \circ}$ be a commutative ring with unity.
Let $x, y \in R$.
Then $x$ and $y$ are associates (in $R$) if and only if there exists a unit $u$ of $\struct {R, +, \circ}$ such that $u \circ x = y$.
Also known as
The statement $x$ is an associate of $y$ can be expressed as $x$ is associated to $y$.
The notation $x \cong y$ is sometimes seen to indicate that $x$ is an associate of $y$.
See, for example, 1949: Helmut Hasse: Zahlentheorie
Also see
- Results about associates can be found here.