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Local duo rings whose finitely generated modules are direct sums of cyclics

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Abstract

In this paper, we give an answer to the following question of Kaplansky [14] in the local case: For which duo rings R is it true that every finitely generated left R-module can be decomposed as a direct sum of cyclic modules? More precisely, we prove that for a local duo ring R, the following are equivalent: (i) Every finitely generated left R-module is a direct sum of cyclic modules; (ii) Every 2-generated left R-module is a direct sum of cyclic modules; (iii) Every factor module of R R ⊕ R is a direct sum of cyclic modules; (iv) Every factor module of R R ⊕ R is serial; (v) Every finitely generated left R-module is serial; (vi) R is uniserial and for every non-zero ideal I of R, R/I is a linearly compact left R-module; (vii) R is uniserial and every indecomposable injective left R-module is left uniserial; and, (viii) Every finitely generated right R-module is a direct sum of cyclic modules.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Isfahan University of Technology, P.O. Box: 84156-83111, Isfahan, Iran

    M. Behboodi & G. Behboodi Eskandari

  2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran

    M. Behboodi

Authors
  1. M. Behboodi
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  2. G. Behboodi Eskandari
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Correspondence to M. Behboodi.

Additional information

The research of the first author was in part supported by a grant from IPM (No. 92130413). This research is partially carried out in the IPM-Isfahan Branch.

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Behboodi, M., Behboodi Eskandari, G. Local duo rings whose finitely generated modules are direct sums of cyclics. Indian J Pure Appl Math 46, 59–72 (2015). https://doi.org/10.1007/s13226-015-0108-9

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  • DOI: https://doi.org/10.1007/s13226-015-0108-9

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