\(H\)-supplemented modules with respect to a preradical

Abstract

Let \(M\) be a right \(R\)-module and \(\tau\) a preradical. We call \(M\) \(\tau\)-\(H\)-supplemented if for every submodule \(A\) of \(M\) there exists a direct summand \(D\) of \(M\) such that \((A + D)/D \subseteq \tau(M/D)\) and \((A + D)/A \subseteq \tau(M/A)\). Let \(\tau\) be a cohereditary preradical. Firstly, for a duo module \(M = M_{1} \oplus M_{2}\) we prove that \(M\) is \(\tau\)-\(H\)-supplemented if and only if \(M_{1}\) and \(M_{2}\) are \(\tau\)-\(H\)-supplemented. Secondly, let \(M=\oplus_{i=1}^nM_i\) be a \(\tau\)-supplemented module. Assume that \(M_i\) is \(\tau\)-\(M_j\)-projective for all \(j > i\). If each \(M_i\) is \(\tau\)-\(H\)-supplemented, then \(M\) is \(\tau\)-\(H\)-supplemented. We also investigate the relations between \(\tau\)-\(H\)-supplemented modules and \(\tau\)-(\(\oplus\)-)supplemented modules.