Editing Commutative ring (section)

=== Localizations ===
{{Main|Localization of a ring}}
The ''localization'' of a ring is a process in which some elements are rendered invertible, i.e. multiplicative inverses are added to the ring. Concretely, if <math> S </math> is a [[multiplicatively closed subset]] of <math> R </math> (i.e. whenever <math> s,t \in S </math> then so is <math> st </math>) then the ''localization'' of <math> R </math> at <math> S </math>, or ''ring of fractions'' with denominators in <math> S </math>, usually denoted <math> S^{-1}R </math> consists of symbols
{{block indent|1= <math>\frac{r}{s}</math> with <math> r \in R, s \in S </math> }}
subject to certain rules that mimic the cancellation familiar from rational numbers. Indeed, in this language <math> \mathbb{Q} </math> is the localization of <math> \mathbb{Z} </math> at all nonzero integers. This construction works for any integral domain <math> R </math> instead of <math> \mathbb{Z} </math>. The localization <math> \left(R\setminus \left\{0\right\}\right)^{-1}R </math> is a field, called the [[quotient field]] of <math> R </math>.