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Direct sums of ADS* modules

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Abstract

A module M is called ADS* if for every direct summand N of M and every supplement K of N in M, we have \(M=N\oplus K\). In this work, we study direct sums of ADS* modules. Many examples are provided to show that this notion is not inherited by direct sums. It is shown that if a module M has a decomposition \(M=A\oplus B\) which complements direct summands such that A and B are mutually projective, then M is ADS*. The class of rings R, for which all direct sums of ADS* R-modules are ADS*, is shown to be exactly that of the right V-rings. We characterize the class of right perfect rings R for which \(R\oplus S\) is ADS* for every simple R-module S as that of the semisimple rings.

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Authors and Affiliations

  1. Centre Régional des Métiers de l’Education et de la Formation (CRMEF)-Tanger, Avenue My Abdelaziz, Souani, BP 3117, Tangier, Morocco

    Rachid Tribak

  2. Department of Mathematics, Hacettepe University, Ankara, 06800, Turkey

    Derya Keskin Tütüncü

  3. Department of Mathematics, Karabük University, Karabük, Turkey

    Nil Orhan Ertaş

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  1. Rachid Tribak
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  2. Derya Keskin Tütüncü
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  3. Nil Orhan Ertaş
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Correspondence to Rachid Tribak.

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Tribak, R., Tütüncü, D.K. & Ertaş, N.O. Direct sums of ADS* modules. Bol. Soc. Mat. Mex. 22, 33–46 (2016). https://doi.org/10.1007/s40590-015-0068-4

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  • DOI: https://doi.org/10.1007/s40590-015-0068-4

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