A note on intersections of valuation ideals

Author:
Charles H. Brase

Journal:
Proc. Amer. Math. Soc. **38** (1973), 37-39

MathSciNet review:
0309915

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Abstract | References | Additional Information

Abstract: In this note it is proved that if is an integral domain the set of valuation ideals of is closed under intersection if and only if the integral closure of is a valuation ring. Let be a domain which is integrally dependent on and contains the integral closure of . Then the set of valuation ideals of is the same as the set of ideals of which contract from ideals of if and only if the set of valuation ideals of is closed under intersection.

**[1]**Robert W. Gilmer Jr.,*Contracted ideals with respect to integral extensions*, Duke Math. J.**34**(1967), 561–571. MR**0218341****[2]**Robert W. Gilmer,*Multiplicative ideal theory*, Queen’s Papers in Pure and Applied Mathematics, No. 12, Queen’s University, Kingston, Ont., 1968. MR**0229624****[3]**Robert Gilmer and Jack Ohm,*Primary ideals and valuation ideals*, Trans. Amer. Math. Soc.**117**(1965), 237–250. MR**0169871**, 10.1090/S0002-9947-1965-0169871-4**[4]**Max. D. Larsen and Paul J. McCarthy,*Multiplicative theory of ideals*, Academic Press, New York-London, 1971. Pure and Applied Mathematics, Vol. 43. MR**0414528****[5]**Oscar Zariski and Pierre Samuel,*Commutative algebra, Volume I*, The University Series in Higher Mathematics, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR**0090581****[6]**-,*Commutative algebra*. Vol. 2, University Ser. in Higher Math., Van Nostrand, Princeton, N.J., 1960. MR**22**#11006.

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1973-0309915-X

Keywords:
Valuation ideal,
completion,
integral closure

Article copyright:
© Copyright 1973
American Mathematical Society