Localization

The process of inverting a specified multiplicative set S in a ring or analogous elements in other algebraic structures to form a new object S^{-1}R in which elements of S become units; more generally, formally adjoining inverses along a chosen class of morphisms in a category.

The construction of a new object by formally adjoining inverses to a specified multiplicative subset, thereby inverting chosen elements while preserving the rest of the structure via a universal property.

The process of adjoining inverses to a chosen multiplicative subset S of an algebraic object (typically a ring or monoid) to form a localized object S^{-1}A in which every element of S becomes invertible; equivalently, the universal construction that makes specified elements units while preserving as much of the original structure as possible.