Direct sums of hollow-lifting modules. (Q2880065)

A module \(M\) is said to be a lifting module if for any submodule \(N\) of \(M\), there exists a direct summand \(K\) of \(M\) such that \(K\subseteq N\) and \(N/K\) is small in \(M/K\). A nonzero module \(M\) is called hollow if every proper submodule of \(M\) is small in \(M\). As a generalization of the notion of lifting modules, hollow-lifting modules were defined by \textit{N. Orhan, D. Keskin Tütüncü} and \textit{R. Tribak}, [in Taiwanese J. Math. 11, No. 2, 545-568 (2007; Zbl 1130.16001)]. A module \(M\) is called hollow-lifting if every submodule \(N\) of \(M\) with \(M/N\) hollow contains a direct summand \(K\) of \(M\) such that \(N/K\) is a small submodule of \(M/K\).NEWLINENEWLINE It is known that a direct sum of hollow-lifting modules need not be hollow-lifting. The paper under review shows that a factor module of a hollow-lifting module need not be hollow-lifting. Some conditions are given under which a direct sum of hollow-lifting modules is hollow-lifting.