a multiplicative set Let be a multiplicative set. The saturation of is the set The multiplicative set is saturated if it equals its saturation, that is, if , or equivalently, if implies that and are in . If is not saturated, and then is a multiplicative inverse of the image of in So, the images of the elements of are all invertible in and the universal property implies that and are canonically isomorphic, that is, there is a unique isomorphism between them that fixes the images of the elements of . If and are two multiplicative sets, then and are isomorphic if and only if they have the same saturation, or, equivalently, if belongs to one of the multiplicative set, then there exists such that belongs to the other. Saturated multiplicative sets are not widely used explicitly, since, for verifying that a set is saturated, one must know all units of the ring. Terminology explained by the context The term localization originates in the general trend of modern mathematics to study geometrical and topological objects locally, that is in terms of their behavior near each point. Examples of this trend are the fundamental concepts of manifolds, germs and sheafs. In algebraic geometry, an affine algebraic set can be identified with a quotient ring of a polynomial ring in such a way that the points of the algebraic set correspond to the maximal ideals of the ring (this is Hilbert's Nullstellensatz). This correspondence has been generalized for making the set of the prime ideals of a commutative ring a topological space equipped with the Zariski topology; this topological space is called the spectrum of the ring. In this context, a localization by a multiplicative set may be viewed as the restriction of the spectrum of a ring to the subspace of the prime ideals (viewed as points) that do not intersect the multiplicative set. Two classes of localizations are more commonly considered: The multiplicative set is the complement of a prime ideal of a ring . In this case, one speaks of the "localization at ", or "localization at a point". The resulting ring, denoted is a local ring, and is the algebraic analog of a ring of germs. The multiplicative set consists of all powers of an element of a ring . The resulting ring is commonly denoted and its spectrum is the Zariski open set of the prime ideals