%0 Journal Article
%T On a generalization of $D3$-modules
%A Diop, Papa Cheikhou 
%A Seye, Modou 
%A Maurya, Sanjeev Kumar 
%A Barry, Mamadou 
%J Indian Journal of Mathematics
%V 67
%N 1
%D 2025
%P 109–132
%@ 0019-5324
%I The Allahabad Mathematical Society
%K Small submodules, cosingular modules, projective modules, $D3$-modules, $D31$-modules.
%X $ R $ is supposed to be an associative ring with identity and $ M $ be a unital left $R$-module. In this paper, we present the concept of a $D31$-module that generalizes the concept of a $D3$-module introduced in  [14] and studied in [25] . A module $M$ is termed as $ D31 $-module if, whenever $ W $ and $ X $ are summands of $ M $ with $ W + X = M $ and $ W $ is cosingular, we have $ W \cap X $ is a direct summand of $ M $. We explore fundamental properties of these modules and establish that the class of rings $ R $ such that every $ D31 $-module is also a $ D3 $-module coincides precisely with the class of COSP-rings. Additionally, we examine the connections between $ D31 $-modules and other related module classes.