# 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**.