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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Almost maximal integral domains and finitely generated modules


Author: Willy Brandal
Journal: Trans. Amer. Math. Soc. 183 (1973), 203-222
MSC: Primary 13G05
MathSciNet review: 0325609
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Abstract: We present a class of integral domains with all finitely generated modules isomorphic to direct sums of cyclic modules. This class contains all previously known examples (i.e., the principal ideal domains and the almost maximal valuation rings) and, by an example, at least one more domain. The class consists of the integral domains satisfying (1) every finitely generated ideal is principal (obviously a necessary condition) and (2) every proper homomorphic image of the domain is linearly compact. We call an integral domain almost maximal if it satisfies (2). This is one of eleven conditions which, for valuation rings, is equivalent of E. Matlis' ``almost maximal.'' An arbitrary integral domain R is almost maximal if and only if it is h-local and $ {R_M}$ is almost maximal for every maximal ideal M of R. Finally, equivalent conditions for a Prüfer domain to be almost maximal are studied, and in the process some conjectures of E. Matlis are answered.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0325609-3
PII: S 0002-9947(1973)0325609-3
Keywords: Finitely generated modules, valuation ring, commutative Noetherian ring, Prüfer domain, Bèzout domain, h-local domain, linearly compact module, algebraically compact module, injective module, R-topology
Article copyright: © Copyright 1973 American Mathematical Society