A homological approach to $\oplus$-supplemented modules

Sepideh Khajvand Sany, Ali Reza Moniri Hamzekolaee, Yahya Talebi

$\oplus$-supplemented modules as a famous generalization of lifting (projective supplemented) modules were widely studied in the last decades. In this paper, we peruse a homological approach to $\oplus$-supplemented modules. Let $R$ be a ring, $M$ a right $R$-module and $S=End_{R}(M)$. We say that $M$ is endomorphism $\oplus$-supplemented (briefly, $E$-$\oplus$-supplemented) provided that for every $f\in S$, there exists a direct summand $D$ of $M$ such that $Imf+D=M$ and $Imf\cap D\ll D$. We investigate some general properties of $E$-$\oplus$-supplemented modules and try to consider their relation with some known classes of modules such as dual Rickart modules, $H$-supplemented modules and $\oplus$-supplemented modules.