Editing Unit (ring theory) (section)

{{Short description|In mathematics, element with a multiplicative inverse}}
{{Distinguish|Unit ring}}
In [[algebra]], a '''unit''' or '''invertible element'''{{efn|In the case of rings, the use of "invertible element" is taken as self-evidently referring to multiplication, since all elements of a ring are invertible for addition.}} of a [[ring (mathematics)|ring]] is an [[invertible element]] for the multiplication of the ring. That is, an element {{mvar|u}} of a ring {{mvar|R}} is a unit if there exists {{mvar|v}} in {{mvar|R}} such that
<math display="block">vu = uv = 1,</math>
where {{math|1}} is the [[multiplicative identity]]; the element {{mvar|v}} is unique for this property and is called the [[multiplicative inverse]] of {{mvar|u}}.{{sfn|Dummit|Foote|2004|ps=}}{{sfn|Lang|2002|ps=}} The set of units of {{mvar|R}} forms a [[Group (mathematics)|group]] {{math|''R''{{sup|×}}}} under multiplication, called the '''group of units''' or '''unit group''' of {{mvar|R}}.{{efn|The notation {{math|''R''{{sup|×}}}}, introduced by [[André Weil]], is commonly used in [[number theory]], where unit groups arise frequently.{{sfn|Weil|1974|ps=}}  The symbol {{math|×}} is a reminder that the group operation is multiplication.  Also, a superscript × is not frequently used in other contexts, whereas a superscript {{math|*}} often denotes dual.}}  Other notations for the unit group are {{math|''R''<sup>∗</sup>}}, {{math|U(''R'')}}, and {{math|E(''R'')}} (from the German term {{lang|de|[[wikt:Einheit|Einheit]]}}).

Less commonly, the term ''unit'' is sometimes used to refer to the element {{math|1}} of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also [[unit matrix]]. Because of this ambiguity, {{math|1}} is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a [[rng (algebra)|rng]].