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Hometotal integral closure

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# 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 $R$.

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 $\mathrm{Quot}(A)$.

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

## Mathematics Subject Classification

13B22*no label found*

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## Comments

## universal property

there is a universal property involved with TIC.. I'll try to add this info once I get the time.