A generalization of ⊕-supplemented modules

Abstract

Let M and X be R-modules. We define the X-⊕-supplemented modules via the classB(M, X) as a generalization of ⊕-supplemented modules. We show that any finite direct sum of X-⊕-supplemented modules is X-⊕-supplemented. It is given a number of necessary and sufficient conditions for every direct summand of an X-⊕-supplemented module to be X-⊕-supplemented.