# Category:Definitions/Associates

This category contains definitions related to Associates in the context of Abstract Algebra.

Related results can be found in Category: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.

## Pages in category "Definitions/Associates"

The following 9 pages are in this category, out of 9 total.

### A

- Definition:Associate
- Definition:Associate in Integral Domain
- Definition:Associate of Integer
- Definition:Associate/Commutative and Unitary Ring
- Definition:Associate/Integers
- Definition:Associate/Integral Domain
- Definition:Associate/Integral Domain/Definition 1
- Definition:Associate/Integral Domain/Definition 2
- Definition:Associate/Integral Domain/Definition 3