On the direct sum of dual-square-free modules

A module M is called square-free if it contains nonon-zero omorphic submodules and B with A∩B= 0. Dually, Mis dual-square-free has no proper M=A+B M/A∼=M/B. In this paper we show that M=⊕i∈I Mi, then iff each Mi Mj ⊕j=i∈I are orthogonal. M=⊕ni=1Mi, dual-square-free, 1⩽i⩽n, ⊕ni=jMi factor-orthogonal. Moreover, in the finite case, weshow M=⊕i∈ISi a direct sum of non-is simple modules, dual-square-free. particular, M=A⊕B where B=⊕i∈ISi ofnon-isomorphic factor-orthogonal; extends an earlier result by theauthors [2, Proposition 2.8].