Bulletin of the
Korean Mathematical Society BKMS

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