Editing Inverse element (section)

==In rings==
A [[ring (mathematics)|ring]] is an [[algebraic structure]] with two operations, ''addition'' and ''multiplication'', which are denoted as the usual operations on numbers. 

Under addition, a ring is an [[abelian group]], which means that addition is [[commutative]] and [[associative]]; it has an identity, called the [[additive identity]], and denoted {{math|0}}; and every element {{mvar|x}} has an inverse, called its [[additive inverse]] and denoted  {{math|−''x''}}. Because of commutativity, the concepts of left and right inverses are meaningless since they do not differ from inverses.

Under multiplication, a ring is a [[monoid]]; this means that multiplication is associative and has an identity called the [[multiplicative identity]] and denoted {{math|1}}. An ''invertible element '' for multiplication is called a [[unit (ring theory)|unit]]. The inverse or [[multiplicative inverse]] (for avoiding confusion with additive inverses) of a unit {{mvar|x}} is denoted <math>x^{-1},</math> or, when the multiplication is commutative, <math display=inline>\frac 1x.</math>

The additive identity {{math|0}} is never a unit, except when the ring is the [[zero ring]], which has {{math|0}} as its unique element.

If {{math|0}} is the only non-unit, the ring is a [[field (mathematics)|field]] if the multiplication is commutative, or a [[division ring]] otherwise.

In a [[noncommutative ring]] (that is, a ring whose multiplication is not commutative), a non-invertible element may have one or several left or right inverses. This is, for example, the case of the [[linear function]]s from an [[infinite-dimensional vector space]] to itself.

A [[commutative ring]] (that is, a ring whose multiplication is commutative) may be extended by adding inverses to elements that are not [[zero divisors]] (that is, their product with a nonzero element cannot be {{math|0}}). This is the process of [[localization (commutative algebra)|localization]], which produces, in particular, the field of [[rational number]]s from the ring of integers, and, more generally, the [[field of fractions]] of an [[integral domain]]. Localization is also used with zero divisors, but, in this case the original ring is not a [[subring]] of the localisation; instead, it is mapped non-injectively to the localization.