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Transactions of the American Mathematical Society

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On $ h$-local integral domains


Author: Willy Brandal
Journal: Trans. Amer. Math. Soc. 206 (1975), 201-212
MSC: Primary 13G05
DOI: https://doi.org/10.1090/S0002-9947-1975-0407004-3
MathSciNet review: 0407004
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Abstract: Related to the question of determining the integral domains with the property that finitely generated modules are a direct sum of cyclic submodules is the question of determining when an integral domain is $ h$-local, especially for Bezout domains. Presented are ten equivalent conditions for a Prüfer domain with two maximal ideals not to be $ h$-local. If $ R$ is an integral domain with quotient field $ Q$, if every maximal ideal of $ R$ is not contained in the union of the rest of the maximal ideals of $ R$, and if $ Q/R$ is an injective $ R$-module, then $ R$ is $ h$-local; and if in addition $ R$ is a Bezout domain, then every finitely generated $ R$-module is a direct sum of cyclic submodules. In particular if $ R$ is a semilocal Prüfer domain with $ Q/R$ an injective $ R$-module, then every finitely generated $ R$-module is a direct sum of cyclic submodules.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0407004-3
Keywords: Integral domain, $ h$-local domain, Bezout domain, Prüfer domain, finitely generated module, injective module
Article copyright: © Copyright 1975 American Mathematical Society

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