Localization (commutative algebra)

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In commutative algebra and algebraic geometry, 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 ring/module R so that it consists of fractions m s , {\displaystyle {\tfrac {m}{s}},} such that the denominator s belongs to a given subset S of R. If S is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field Q {\displaystyle \mathbb {Q} } of rational numbers from the ring Z {\displaystyle \mathbb {Z} } of integers. The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory...

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