total integral closure

A commutative unitary ring $R$ is said to be totally integrally closed 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 (http://planetmath.org/ReducedRing).

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

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

Let $A$ be an integral domain, then its total integral closure is the integral closure of $A$ in the algebraic closure of $\mathrm{Quot}(A)$.

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