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

Author

Listed:
  • M. Behboodi

    (Isfahan University of Technology
    Institute for Research in Fundamental Sciences (IPM))

  • G. Behboodi Eskandari

    (Isfahan University of Technology)

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.

Suggested Citation

  • M. Behboodi & G. Behboodi Eskandari, 2015. "Local duo rings whose finitely generated modules are direct sums of cyclics," Indian Journal of Pure and Applied Mathematics, Springer, vol. 46(1), pages 59-72, February.
    • Handle: RePEc:spr:indpam:v:46:y:2015:i:1:d:10.1007_s13226-015-0108-9
    DOI: 10.1007/s13226-015-0108-9
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    Cited by:

    1. Ulrich Albrecht & Francisco Javier Santillán-Covarrubias, 2021. "Hopficity and duo rings," Indian Journal of Pure and Applied Mathematics, Springer, vol. 52(2), pages 369-374, June.

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