Editing Inverse element (section)

{{Short description|Generalization of additive and multiplicative inverses}}
{{redirect|Invertible|other uses|Invertible (disambiguation)}}

In [[mathematics]], the concept of an '''inverse element''' generalises the concepts of [[additive inverse|opposite]] ({{math|−''x''}}) and [[Multiplicative inverse|reciprocal]] ({{math|1/''x''}}) of numbers. 

Given an [[operation (mathematics)|operation]] denoted here {{math|∗}}, and an [[identity element]] denoted {{mvar|e}}, if {{math|1=''x'' ∗ ''y'' = ''e''}}, one says that {{mvar|x}} is a '''left inverse''' of {{mvar|y}}, and that {{mvar|y}} is a '''right inverse''' of {{mvar|x}}. (An identity element is an element such that {{math|1=''x'' * ''e'' = ''x''}} and {{math|1=''e'' * ''y'' = ''y''}} for all {{mvar|x}} and {{mvar|y}} for which the left-hand sides are defined.<ref>The usual definition of an identity element has been generalized for including the [[identity function]]s as identity elements for [[function composition]], and [[identity matrices]] as identity elements for [[matrix multiplication]].</ref>)

When the operation {{math|∗}} is [[associative]], if an element {{mvar|x}} has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the ''inverse element'' or simply the ''inverse''. Often an adjective is added for specifying the operation, such as in [[additive inverse]], [[multiplicative inverse]], and [[functional inverse]]. In this case (associative operation), an  '''invertible element''' is an element that has an inverse. In a [[ring (mathematics)|ring]], an ''invertible element'', also called a [[unit (ring theory)|unit]], is an element that is invertible under multiplication (this is not ambiguous, as every element is invertible under addition).

Inverses are commonly used in [[group (mathematics)|groups]]{{mdash}}where every element is invertible, and [[ring (mathematics)|rings]]{{mdash}}where invertible elements are also called [[unit (ring theory)|units]]. They are also commonly used for operations that are not defined for all possible operands, such as [[inverse matrices]] and [[inverse function]]s. This has been generalized to [[category theory]], where, by definition, an [[isomorphism]] is an invertible [[morphism]].

The word 'inverse' is derived from {{langx|la|[[wikt:inversus|inversus]]}} that means 'turned upside down', 'overturned'. This may take its origin from the case of [[fraction (mathematics)|fractions]], where the (multiplicative) inverse is obtained by exchanging the numerator and the denominator (the inverse of <math>\tfrac x y</math> is <math>\tfrac y x</math>).

{{hatnote|In this article, the operations are [[associative]] and have [[identity element]]s, except when otherwise stated and in section {{slink||Generalizations}}.}}