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.