Commutative ring
Localizations

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 S {\displaystyle S} is a multiplicatively closed subset of R {\displaystyle R} (i.e. whenever s, t ∈ S {\displaystyle s,t\in S} then so is s t {\displaystyle st}) then the localization of R {\displaystyle R} at S {\displaystyle S}, or ring of fractions with denominators in S {\displaystyle S}, usually denoted S − 1 R {\displaystyle S^{-1}R} consists of symbols

subject to certain rules that mimic the cancellation familiar from rational numbers. Indeed, in this language Q {\displaystyle \mathbb {Q} } is the localization of Z {\displaystyle \mathbb {Z} } at all nonzero integers. This construction works for any integral domain R {\displaystyle R} instead of Z {\displaystyle \mathbb {Z} }. The localization (R ∖ { 0 }) − 1 R {\displaystyle \left(R\setminus \left\{0\right\}\right)^{-1}R} is a field, called the quotient field of R {\displaystyle R}.
keywords:letter_frequency