integrity characterized by places


Let R be a subring of the field K,  1R.  An element α of the field is integral over R if and only if all places ( φ of K satisfy the implicationMathworldPlanetmath


1.  Let R be a subring of the field K,  1R.  The integral closureMathworldPlanetmath of R in K is the intersectionMathworldPlanetmath of all valuation domains in K which contain the ring R.  The integral closure is integrally closedMathworldPlanetmath in the field K.

2.  Every valuation domain is integrally closed in its field of fractionsMathworldPlanetmath.

Title integrity characterized by places
Canonical name IntegrityCharacterizedByPlaces
Date of creation 2013-03-22 14:56:57
Last modified on 2013-03-22 14:56:57
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Theorem
Classification msc 12E99
Classification msc 13B21
Related topic Integral
Related topic PlaceOfField