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 $\\bar{R}$ 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.

单词	TotalIntegralClosure
释义	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 $\\bar{R}$ 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.
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