often the algebraic counterpart of geometric local properties of algebraic varieties, which are properties that can be studied by restriction to a small neighborhood of each point of the variety. (There is another concept of local property that refers to localization to Zariski open sets; see , below.) Many local properties are a consequence of the fact that the module is a faithfully flat module when the direct sum is taken over all prime ideals (or over all maximal ideals of ). See also Faithfully flat descent. Examples of local properties A property of an -module is a local property if the following conditions are equivalent: holds for . holds for all where is a prime ideal of . holds for all where is a maximal ideal of . The following are local properties: is zero. is torsion-free (in the case where is a commutative domain). is a flat module. is an invertible module (in the case where is a commutative domain, and is a submodule of the field of fractions of ). is injective (resp. surjective), where is another -module. On the other hand, some properties are not local properties. For example, an infinite direct product of fields is not an integral domain nor a Noetherian ring, while all its local rings are fields, and therefore Noetherian integral domains. Localization to Zariski open sets Non-commutative case Localizing non-commutative rings is more difficult. While the localization exists for every set S of prospective units, it might take a different form to the one described above. One condition which ensures that the localization is well behaved is the Ore condition. One case for non-commutative rings where localization has a clear interest is for rings of differential operators. It has the interpretation, for example, of adjoining a formal inverse D−1 for a differentiation operator D. This is done in many contexts in methods for differential equations. There is now a large mathematical theory about it, named microlocalization, connecting with numerous other branches. The micro- tag is to do with connections with Fourier theory, in particular. See also Local analysis Localization of a category Localization of a topological space References Atiyah and MacDonald. Introduction to Commutative Algebra. Addison-Wesley. Borel, Armand. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. . Matsumura. Commutative Algebra. Benjamin-Cummings Serge Lang, "Algebraic Number Theory," Springer, 2000. pages 3–4. External links Localization from MathWorld. Ring theory Module theory Localization (mathematics)