total integral closure

A commutativePlanetmathPlanetmathPlanetmath unitary ring R is said to be totally integrally closedMathworldPlanetmath if it does not have an overring which is both an integral and an essential extension of R.

All totally integrally closed rings are reduced (

Suppose that R is any commutative ring and that R¯ is an integral and essential extension of R. If R¯ is a totally integrally closed ring, then R¯ is called a total integral closure of R.

For fields the concept totally integrally closed, integrally closed and algebraically closedMathworldPlanetmath coincide.

Let A be an integral domainMathworldPlanetmath, then its total integral closure is the integral closureMathworldPlanetmath of A in the algebraic closure of Quot(A).

Enochs has first proven that all commutative reduced rings have total integral closure and this is unique up to ring isomorphism.

Title total integral closure
Canonical name TotalIntegralClosure
Date of creation 2013-03-22 18:51:44
Last modified on 2013-03-22 18:51:44
Owner jocaps (12118)
Last modified by jocaps (12118)
Numerical id 14
Author jocaps (12118)
Entry type Definition
Classification msc 13B22
Defines total integral closure