Associate elements

Builds on Unit of a ring Divisibility in a commutative ring

Definition

Let $R$ be a commutative ring and let $a, b \in R$ . The elements $a$ and $b$ are associates, written $a \sim b$ , if there is a unit $u \in R$ with $b = u a$ .

Remarks

Being associates is an equivalence relation. In an integral domain, $a \sim b$ 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 $1 - i$ .

Reading path

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Group concept
Ring definition
Commutative ring definition
Unit of a ring definition
Divisibility in a commutative ring definition
Associate elements definition

Where this leads

31 pages build on this one.

Tag	`053E86NWVK`
URI	`m2://053E86NWVK`
Kind	definition
Writing	W0 stub
MSC	13A05
Defines	Associate elements
Uses	Unit of a ring, Divisibility in a commutative ring

Associate elements

Definition

Remarks

Reading path

Where this leads

Further reading

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