Decomposability of finitely presented modules

Warfield, R. B.

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

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.

Gorô Azumaya, Corrections and supplementaries to my paper concerning Krull-Remak-Schmidt’s theorem, Nagoya Math. J. 1 (1950), 117–124. MR 37832
D. G. Higman, Indecomposable representations at characteristic $p$, Duke Math. J. 21 (1954), 377–381. MR 67896
Irving Kaplansky, Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc. 72 (1952), 327–340. MR 46349, DOI https://doi.org/10.1090/S0002-9947-1952-0046349-0

W. Krull, Die verschiedenen Arten der Hauptidealringe, S.-B. Heidelberger Akad. Wiss. 6 (1924).

A. I. Uzkov, On the decomposition of modules over a commutative ring into direct sums of cyclic submodules, Mat. Sb. (N.S.) 62(104) (1963), 469–475 (Russian). MR 0157986
R. B. Warfield Jr., Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969), 699–719. MR 242885
Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581