extension by localization

Let R be a commutative ring and let S be a non-empty multiplicative subset of R.  Then the localisation (http://planetmath.org/LocalizationMathworldPlanetmath) of R at S gives the commutative ring  S-1R  but, generally, it has no subring isomorphicPlanetmathPlanetmathPlanetmath to R.  Formally, S-1R consists of all elements as (aR, sS).  Therefore, S-1R is called also a ring of quotients of R.  If  0S, then  S-1R={0};  we assume now that  0S.

  • The mappingaass, where s is any element of S, is well-defined and a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from R to S-1R.  All elements of S are mapped to units of S-1R.

  • If, especially, S contains no zero divisorsMathworldPlanetmath of the ring R, then the above mapping is an isomorphism from R to a certain subring of S-1R, and we may think that  S-1RR.  In this case, the ring of fractions of R is an extension ring of R; this concerns of course the case that R is an integral domainMathworldPlanetmath.  But if R is a finite ring, then  S-1R=R,  and no proper extension is obtained.

Title extension by localization
Canonical name ExtensionByLocalization
Date of creation 2013-03-22 14:24:42
Last modified on 2013-03-22 14:24:42
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 15
Author pahio (2872)
Entry type Definition
Classification msc 13B30
Synonym ring extension by localization
Related topic TotalRingOfFractions
Related topic ClassicalRingOfQuotients
Related topic FiniteRingHasNoProperOverrings
Defines ring of fractions
Defines ring of quotients