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# integrity characterized by places

###### Theorem.

Corollary 1. Let $R$ be a subring of the field $K$, $1\in R$. The integral closure of $R$ in $K$ is the intersection of all valuation domains in $K$ which contain the ring $R$. The integral closure is integrally closed in the field $K$.

Corollary 2. Every valuation domain is integrally closed in its field of fractions.

Keywords:

integral over a ring

Related:

Integral, PlaceOfField

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

12E99*no label found*13B21

*no label found*

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new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag