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.

Suppose that $R$ is any commutative ring and that $\bar R$ is an integral and essential extension of $R$ . If $\bar R$ is a totally integrally closed ring, then $\bar R$ 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 $\qf A$ .

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