• Honam Mathematical Journal
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• v.46 no.4
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• pp.650-656
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• 2024

In this work ⊕ - e-supplemented modules are defined and some properties of these modules are investigated. It is proved that the finite direct sum of ⊕-e-supplemented modules is also ⊕-e-supplemented. Let M be a distributive and ⊕-e-supplemented R-module. Then every factor module and homomorphic image of M are ⊕ - e-supplemented. Let M be a ⊕ - e-supplemented R-module with SSP property. Then every direct summand of M is ⊕ - e-supplemented.

• 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 M_R 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.