Addition and scalar multiplication are defined as for usual fractions (in the following formula, and ): Moreover, is also an -module with scalar multiplication It is straightforward to check that these operations are well-defined, that is, they give the same result for different choices of representatives of fractions. The localization of a module can be equivalently defined by using tensor products: The proof of equivalence (up to a canonical isomorphism) can be done by showing that the two definitions satisfy the same universal property. Module properties If is a submodule of an -module , and is a multiplicative set in , one has This implies that, if is an injective module homomorphism, then is also an injective homomorphism. Since the tensor product is a right exact functor, this implies that localization by maps exact sequences of -modules to exact sequences of -modules. In other words, localization is an exact functor, and is a flat -module. This flatness and the fact that localization solves a universal property make that localization preserves many properties of modules and rings, and is compatible with solutions of other universal properties. For example, the natural map is an isomorphism. If is a finitely presented module, the natural map is also an isomorphism. If a module M is a finitely generated over R, one has where denotes annihilator, that is the ideal of the elements of the ring that map to zero all elements of the module. In particular, that is, if for some Localization at primes The definition of a prime ideal implies immediately that the complement of a prime ideal in a commutative ring is a multiplicative set. In this case, the localization is commonly denoted The ring is a local ring, that is called the local ring of at This means that is the unique maximal ideal of the ring Such localizations are fundamental for commutative algebra and algebraic geometry for several reasons. One is that local rings are often easier to study than general commutative rings, in particular because of Nakayama lemma. However, the main reason is that many properties are true for a ring if and only if they are true for all its local rings. For example, a ring is regular if and only if all its local rings are regular local rings. Properties of a ring that can be characterized on its local rings are called local properties, and are