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A note on valuations associated with a local domain

Published online by Cambridge University Press:  24 October 2008

D. Rees
D. Rees
Affiliation:
Downing CollegeCambridge
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Extract

The purpose of the present note is to prove the following two theorems:

Theorem 1. Let Q be an equicharacteristic local domain with maximal ideal m. Let a be any ideal of Q. Then the intersection of all integrally closed m-primary ideals of Q which contain a is the integral closure ā of a.

Theorem 2. If Q is as above, and if S denotes the set of valuations on the field of fractions F of Q which are associated with Q, then the intersection of the valuation rings belonging to valuations in S is the integral closure of Q.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 51 , Issue 2 , April 1955 , pp. 252 - 253
Copyright
Copyright © Cambridge Philosophical Society 1955

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References

REFERENCES

(1)Rees, D.Valuations associated with local rings. I. Submitted to Proc. Lond. math. Soc.Google Scholar
(2)Weil, A.Arithmetic on algebraic varieties. Ann. Math., Princeton, (2), 53 (1951), 411–44.CrossRefGoogle Scholar