that do not contain . Thus the localization is the analog of the restriction of a topological space to a neighborhood of a point (every prime ideal has a neighborhood basis consisting of Zariski open sets of this form). In number theory and algebraic topology, when working over the ring of the integers, one refers to a property relative to an integer as a property true at or away from , depending on the localization that is considered. "Away from " means that the property is considered after localization by the powers of , and, if is a prime number, "at " means that the property is considered after localization at the prime ideal . This terminology can be explained by the fact that, if is prime, the nonzero prime ideals of the localization of are either the singleton set or its complement in the set of prime numbers. Localization and saturation of ideals Let be a multiplicative set in a commutative ring , and be the canonical ring homomorphism. Given an ideal in , let the set of the fractions in whose numerator is in . This is an ideal of which is generated by , and called the localization of by . The saturation of by is it is an ideal of , which can also defined as the set of the elements such that there exists with Many properties of ideals are either preserved by saturation and localization, or can be characterized by simpler properties of localization and saturation. In what follows, is a multiplicative set in a ring , and and are ideals of ; the saturation of an ideal by a multiplicative set is denoted or, when the multiplicative set is clear from the context, (this is not always true for strict inclusions) If is a prime ideal such that then is a prime ideal and ; if the intersection is nonempty, then and Localization of a module Let be a commutative ring, be a multiplicative set in , and be an -module. The localization of the module by , denoted , is an -module that is constructed exactly as the localization of , except that the numerators of the fractions belong to . That is, as a set, it consists of equivalence classes, denoted , of pairs , where and and two pairs and are equivalent if there is an element in such that