Editing Ring (mathematics) (section)

=== Valuation ring ===
{{main|Valuation ring}}

If {{mvar|K}} is a field, a [[valuation (algebra)|valuation]] {{mvar|v}} is a group homomorphism from the multiplicative group {{math|''K''{{sup|∗}}}} to a totally ordered abelian group {{mvar|G}} such that, for any {{math|''f''}}, {{math|''g''}} in {{mvar|K}} with {{math|''f'' + ''g''}} nonzero, {{math|''v''(''f'' + ''g'') ≥ min{''v''(''f''), ''v''(''g'')}.}} The [[valuation ring]] of {{mvar|v}} is the subring of {{mvar|K}} consisting of zero and all nonzero {{mvar|f}} such that {{math|''v''(''f'') ≥ 0}}.

Examples:
* The field of [[formal Laurent series]] <math>k(\!(t)\!)</math> over a field {{mvar|k}} comes with the valuation {{mvar|v}} such that {{math|''v''(''f'')}} is the least degree of a nonzero term in {{mvar|f}}; the valuation ring of {{mvar|v}} is the [[formal power series ring]] <math>k[\![t]\!].</math>
* More generally, given a field {{mvar|k}} and a totally ordered abelian group {{mvar|G}}, let <math>k(\!(G)\!)</math> be the set of all functions from {{mvar|G}} to {{mvar|k}} whose supports (the sets of points at which the functions are nonzero) are [[well ordered]]. It is a field with the multiplication given by [[convolution]]: <math display="block">(f*g)(t) = \sum_{s \in G} f(s)g(t - s).</math> It also comes with the valuation {{mvar|v}} such that {{math|''v''(''f'')}} is the least element in the support of {{mvar|f}}. The subring consisting of elements with finite support is called the [[group ring]] of {{mvar|G}} (which makes sense even if {{mvar|G}} is not commutative). If {{mvar|G}} is the ring of integers, then we recover the previous example (by identifying {{mvar|f}} with the series whose {{mvar|n}}th coefficient is&nbsp;{{math|''f''(''n'')}}.)

{{See also|Novikov ring|uniserial ring}}