Weakly $\oplus$-supplemented modules and weakly $D2$ modules
Bull. Korean Math. Soc. 2020 Vol. 57, No. 3, 691-707
Published online May 7, 2020
Printed May 31, 2020
Phan The Hai, Muhammet Tamer Ko\c{s}an, Truong Cong Quynh
Ton Duc Thang University; Gazi University; The University of Danang - University of Science and Education
Abstract : In this paper, we introduce and study the notions of weakly $\oplus$-supplemented modules, weakly $D2$ modules and weakly $D2$-covers. A right $R$-module $M$ is called weakly $\oplus$-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 $s\in S$ and $s\ne 0$, if there exists $n\in \N$ such that $s^n\ne 0$ and ${\rm Im} (s^n)$ is a direct summand of $M$, then ${\rm Ker} (s^n)$ is a direct summand of $M$. The class of weakly $\oplus$-supplemented-modules and weakly $D2$ modules contains $\oplus$-supplemented modules and $D2$ modules, respectively, and they are equivalent in case $M$ is uniform, and projective, respectively.
Keywords : (weakly) $\oplus$-supplemented module, (weakly) $D2$ module, $GD2$ module, supplement submodule, semiperfect ring, projective cover.
MSC numbers : 16D40, 16D80, 16N80
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