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In commutative algebra and algebraic geometry, the localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing one so that it consists of fractions such that the denominator s belongs to a given subset S of R. The basic example is the construction of the ring Q of rational numbers from the ring Z of rational integers. The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, the term localization originates in algebraic geometry: if R is a ring of functions defined on some geometric object (algebraic variety) V, and one wants to study this variety "locally" near a point p, then one considers the set S of all functions which are not zero at p and localizes R with respect to S. The resulting ring R* contains only information about the behavior of V near p. Cf. the example given at local ring. An important related process is completion: one often localizes a ring/module, then completes. In this article, a ring is commutative with unity.