# Definition:Associate/Integral Domain

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## 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.

## Also known as

The statement **$x$ is an associate of $y$** can be expressed as **$x$ is associated to $y$**.

## Also see

- Results about
**associates**can be found here.