Let R be a commutative ring and let S be a nonempty multiplicative subset of R. The localizationMathworldPlanetmath of R at S is the ring S-1R whose elements are equivalence classesMathworldPlanetmathPlanetmath of R×S under the equivalence relation (a,s)(b,t) if r(at-bs)=0 for some rS. AdditionPlanetmathPlanetmath and multiplication in S-1R are defined by:

  • (a,s)+(b,t)=(at+bs,st)

  • (a,s)(b,t)=(ab,st)

The equivalence class of (a,s) in S-1R is usually denoted a/s. For aR, the localization of R at the minimalPlanetmathPlanetmath multiplicative set containing a is written as Ra. When S is the complementPlanetmathPlanetmath of a prime idealMathworldPlanetmathPlanetmathPlanetmath 𝔭 in R, the localization of R at S is written R𝔭.

Title localization
Canonical name Localization
Date of creation 2013-03-22 11:50:21
Last modified on 2013-03-22 11:50:21
Owner djao (24)
Last modified by djao (24)
Numerical id 11
Author djao (24)
Entry type Definition
Classification msc 13B30
Synonym ring of fractions
Related topic FractionField