• Title/Summary/Keyword: supplement submodule

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SA-SUPPLEMENT SUBMODULES

  • Durgun, Yilmaz
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.1
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    • pp.147-161
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    • 2021
  • In this paper, we introduced and studied sa-supplement submodules. A submodule U of a module V is called an sa-supplement submodule in V if there exists a submodule T of V such that V = T + U and U ∩ T is semiartinian. The class of sa-supplement sequences ������ is a proper class which is generated by socle-free modules injectively. We studied modules that have an sa-supplement in every extension, modules whose all submodules are sa-supplement and modules whose all sa-supplement submodules are direct summand. We provided new characterizations of right semiartinian rings and right SSI rings.

On Lifting Modules and Weak Lifting Modules

  • Tutuncu, Derya Keskin;Tribak, Rachid
    • Kyungpook Mathematical Journal
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    • v.45 no.3
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    • pp.445-453
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    • 2005
  • We say that a module M is weak lifting if M is supplemented and every supplement submodule of M is a direct summand. The module M is called lifting, if it is weak lifting and amply supplemented. This paper investigates the structure of weak lifting modules and lifting modules having small radical over commutative noetherian rings.

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WEAKLY ⊕-SUPPLEMENTED MODULES AND WEAKLY D2 MODULES

  • Hai, Phan The;Kosan, Muhammet Tamer;Quynh, Truong Cong
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.3
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    • pp.691-707
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    • 2020
  • 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.