Editing Inverse element (section)

==In monoids==
A [[monoid]] is a set with an [[associative operation]] that has an [[identity element]].

The ''invertible elements'' in a monoid form a [[group (mathematics)|group]] under monoid operation.

A [[ring (mathematics)|ring]] is a monoid for ring multiplication. In this case, the invertible elements are also called [[unit (ring theory)|units]] and form the [[group of units]] of the ring.

If a monoid is not [[commutative monoid|commutative]], there may exist non-invertible elements that have a left inverse or a right inverse (not both, as, otherwise, the element would be invertible). 

For example, the set of the [[function (mathematics)|functions]] from a set to itself is a monoid under [[function composition]]. In this monoid, the invertible elements are the [[bijective function]]s; the elements that have left inverses are the [[injective function]]s, and those that have right inverses are the [[surjective function]]s.

Given a monoid, one may want extend it by adding inverse to some elements. This is generally impossible for non-commutative monoids, but, in a commutative monoid, it is possible to add inverses to the elements that have the [[cancellation property]] (an element {{mvar|x}} has the cancellation property if <math>xy=xz</math> implies <math>y=z,</math> and <math>yx=zx</math> implies {{nowrap|1=<math>y=z</math>).}} This extension of a monoid is allowed by [[Grothendieck group]] construction. This is the method that is commonly used for constructing [[integer]]s from [[natural number]]s, [[rational number]]s from [[integer]]s and, more generally, the [[field of fractions]] of an [[integral domain]], and [[localization (commutative algebra)|localizations]] of [[commutative ring]]s.