• 호남수학학술지
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• 제41권3호
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• pp.531-538
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• 2019

We introduce FI-${\oplus}$-supplemented modules as a proper generalization of ${\oplus}$-supplemented modules. We prove that; (1) every finite direct sum of FI-${\oplus}$-supplemented R-modules is an FI-${\oplus}$-supplemented R-module for any ring R ; (2) if every left R-module is FI-${\oplus}$-supplemented over a semilocal ring R, then R is left perfect; (3) if M is a finitely generated torsion-free uniform R-module over a commutative integrally closed domain such that every direct summand of M is FI-${\oplus}$-supplemented, then M is a direct sum of cyclic modules.

• Kyungpook Mathematical Journal
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• 제53권1호
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• pp.37-47
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• 2013

Let M and X be right R-modules. We introduce several modules relative to the class of B(M, X) and we investigate relation among these modules. In this note, we show if M is X-${\oplus}$-supplemented such that $M=M_1{\oplus}M_2$ implies $M_1$ and $M_2$ are relatively B-projective, then M is an X-H-supplemented module.

• 대한수학회보
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• 제57권3호
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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.