On the direct sum of dual-square-free modules

Abstract

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