Editing Unit (ring theory) (section)

== Associatedness ==
Suppose that {{mvar|R}} is commutative.  Elements {{mvar|r}} and {{mvar|s}} of {{mvar|R}} are called ''{{visible anchor|associate}}'' if there exists a unit {{mvar|u}} in {{mvar|R}} such that {{math|1=''r'' = ''us''}}; then write {{math|''r'' ~ ''s''}}.  In any ring, pairs of [[additive inverse]] elements{{efn|{{mvar|x}} and {{math|−''x''}} are not necessarily distinct.  For example, in the ring of integers modulo 6, one has {{math|1=3 = −3}} even though {{math|1 ≠ −1}}.}} {{math|''x''}} and {{math|−''x''}} are [[Associated element|associate]], since any ring includes the unit {{math|−1}}.  For example, 6 and −6 are associate in {{math|'''Z'''}}.  In general, {{math|~}} is an [[equivalence relation]] on {{mvar|R}}.

Associatedness can also be described in terms of the [[Group action (mathematics)|action]] of {{math|''R''{{sup|×}}}} on {{mvar|R}} via multiplication: Two elements of {{mvar|R}} are associate if they are in the same {{math|''R''{{sup|×}}}}-[[orbit (group theory)|orbit]].

In an [[integral domain]], the set of associates of a given nonzero element has the same [[cardinality]] as {{math|''R''{{sup|×}}}}.

The equivalence relation {{math|~}} can be viewed as any one of [[Green's relations|Green's semigroup relations]] specialized to the multiplicative [[semigroup]] of a commutative ring {{mvar|R}}.