On a question concerning D4-modules

Abstract

An R-module M is called a D4-module if ‘whenever M1 and M2 are direct summands of M with M1 + M2 = M and M1 ∼= M2, then M1 ∩ M2 is a direct summand of M’. Let M = ⊕i∈IMi be a direct sum of submodules Mi with Hom(Mi,Mj ) = 0 for distinct i, j ∈ I. We show that M is a D4-module if and only if for each i ∈ I the module Mi is a D4-module. This settles an open question concerning direct sums of D4-modules. Our approach is independent of the solution obtained by D’Este, Keskin Tütüncü and Tribak recently.