Decomposability of finitely presented modules

Warfield, R. B.

doi:10.1090/S0002-9939-1970-0254030-4

Decomposability of finitely presented modules
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by R. B. Warfield

Proc. Amer. Math. Soc. 25 (1970), 167-172

DOI: https://doi.org/10.1090/S0002-9939-1970-0254030-4

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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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Irving Kaplansky, Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc. 72 (1952), 327–340. MR 46349, DOI 10.1090/S0002-9947-1952-0046349-0

Die verschiedenen Arten der Hauptidealringe

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