 The ring formula_176 is called the localization of "R" with respect to "S".For example, if "R" is a commutative ring and "f" an element in "R", then the localization formula_179 consists of elements of the form formula_180 (to be precise, formula_181)

The localization is frequently applied to a commutative ring "R" with respect to the complement of a prime ideal (or a union of prime ideals) in "R".In that case formula_182, one often writes formula_183 for formula_176.