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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 College Cambridge
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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.

Information

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.Google Scholar