Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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WEAKLY ⊕-SUPPLEMENTED MODULES AND WEAKLY D2 MODULES

  • Hai, Phan The (Department for Management of Science and Technology Development Ton Duc Thang University) ;
  • Kosan, Muhammet Tamer (Department of Mathematics Gazi University) ;
  • Quynh, Truong Cong (The University of Danang - University of Science and Education)
  • Received : 2019.04.22
  • Accepted : 2020.03.16
  • Published : 2020.05.31
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Abstract

In this paper, we introduce and study the notions of weakly ⊕-supplemented modules, weakly D2 modules and weakly D2-covers. A right R-module M is called weakly ⊕-supplemented if every non-small submodule of M has a supplement that is not essential in M, and module MR is called weakly D2 if it satisfies the condition: for every s ∈ S and s ≠ 0, if there exists n ∈ ℕ such that sn ≠ 0 and Im(sn) is a direct summand of M, then Ker(sn) is a direct summand of M. The class of weakly ⊕-supplemented-modules and weakly D2 modules contains ⊕-supplemented modules and D2 modules, respectively, and they are equivalent in case M is uniform, and projective, respectively.

Keywords

Acknowledgement

The authors wish to thank the referee for his/her comments which helped to improve the present version of this article. The authors also wish to thank Rachid Tribak for Example 2.3.

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