# 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.