---
tag: 053E86NWVK
kind: definition
name: Associate elements
msc: [13A05]
defines: 02879QH7BY
uses: [019NFGFPR5, 04GNK7K4VP]
sources:
  - url: https://en.wikipedia.org/wiki/Integral_domain
    locator: Divisibility, prime elements, and irreducible elements
---

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $a, b \in R$.
The elements $a$ and $b$ are associates, written $a \sim b$, if there is a [[unit|019NFGFPR5]] $u \in R$ with $b = u a$.

## Remarks

Being associates is an equivalence relation.
In an [[integral domain|09S3PFBEPM]], $a \sim b$ if and only if $a \mid b$ and $b \mid a$, if and only if $(a) = (b)$.
In $\mathbb{Z}$, the associates of $n$ are $n$ and $-n$.
In $\mathbb{Z}[i]$, the associates of $1 + i$ are $1 + i$, $-1 - i$, $-1 + i$ and $1 - i$.

## Explanations

### L1 human

Associates are elements that differ only by a unit factor.
For divisibility they behave in the same way.
In the integers, $6$ and $-6$ are associates.