Valuation rings with zero divisors
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- by Patrick H. Kelly and Max D. Larsen PDF
- Proc. Amer. Math. Soc. 30 (1971), 426-430 Request permission
Abstract:
Manis has developed a valuation theory for commutative rings which extends valuation theory for fields. However his results do not extend the characterization of valuation rings as domains of maximal partial homomorphisms. In this note we show that Manis’ theory also generalizes this aspect of valuation theory. Next we show that an overring W of a valuation ring V is not necessarily a valuation ring in any nice sense, but that W is contained in a valuation ring which is a large quotient ring of V.References
- Monte B. Boisen Jr. and Max D. Larsen, Prüfer and valuation rings with zero divisors, Pacific J. Math. 40 (1972), 7–12. MR 309921
- N. Bourbaki, Éléments de mathématique. Fasc. XXX. Algèbre commutative. Chapitre 5: Entiers. Chapitre 6: Valuations, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1308, Hermann, Paris, 1964 (French). MR 0194450
- Malcolm Griffin, Prüfer rings with zero divisors, J. Reine Angew. Math. 239(240) (1969), 55–67. MR 255527, DOI 10.1515/crll.1969.239-240.55
- D. K. Harrison, Finite and infinite primes for rings and fields, Mem. Amer. Math. Soc. 68 (1966), 62. MR 207735
- Merle E. Manis, Valuations on a commutative ring, Proc. Amer. Math. Soc. 20 (1969), 193–198. MR 233813, DOI 10.1090/S0002-9939-1969-0233813-2
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 30 (1971), 426-430
- MSC: Primary 13.98
- DOI: https://doi.org/10.1090/S0002-9939-1971-0285527-X
- MathSciNet review: 0285527