Reductions of ideals in Prüfer domains
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- by James H. Hays PDF
- Proc. Amer. Math. Soc. 52 (1975), 81-84 Request permission
Abstract:
All rings under consideration are Prüfer domains or valuation domains. We characterize the set of basic ideals and the set of $C$-ideals in an arbitrary valuation ring. Basic ideals were introduced in 1954 by Northcott and Rees. The concept of a $C$-ideal is, in a sense, directly opposite to that of a basic ideal. We then prove that a necessary and sufficient condition for every ideal in a domain $D$ to be basic is that $D$ be a one-dimensional Prüfer domain.References
- Robert W. Gilmer, Multiplicative ideal theory, Queen’s Papers in Pure and Applied Mathematics, No. 12, Queen’s University, Kingston, Ont., 1968. MR 0229624
- James H. Hays, Reductions of ideals in commutative rings, Trans. Amer. Math. Soc. 177 (1973), 51–63. MR 323770, DOI 10.1090/S0002-9947-1973-0323770-8
- D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145–158. MR 59889, DOI 10.1017/s0305004100029194
- Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 52 (1975), 81-84
- MSC: Primary 13F05
- DOI: https://doi.org/10.1090/S0002-9939-1975-0376655-2
- MathSciNet review: 0376655