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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Decomposability of finitely presented modules


Author: R. B. Warfield
Journal: Proc. Amer. Math. Soc. 25 (1970), 167-172
MSC: Primary 13.40
DOI: https://doi.org/10.1090/S0002-9939-1970-0254030-4
MathSciNet review: 0254030
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Abstract: It is proved that a commutative ring with $1$ has the property that every finitely presented module is a summand of a direct sum of cyclic modules if and only if it is locally a generalized valuation ring. A Noetherian ring has this property if and only if it is a direct product of a finite number of Dedekind domains and an Artinian principal ideal ring. Any commutative local ring which is not a generalized valuation ring has finitely presented indecomposable modules requiring arbitrarily large numbers of generators.


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Keywords: Indecomposable modules, generalized valuation rings, direct sums of cyclic modules
Article copyright: © Copyright 1970 American Mathematical Society