A generalization of the class of principally lifting modules (Q2409611)

For right \(R\)-modules, with \(R\) an arbitrary ring, the concept of \(\delta\)-small modules was introduced by \textit{Y. Zhou} [Alg. Colloq. 7, 305--318 (2000; Zbl. 0994.16016)]. \textit{G.F. Birkenmeier} et al. [Glasgow Math. J. 52, 41--52 (2010; Zbl. 1215.16006)] defined an equivalence relation \(\beta\) involving the concept of smallness on submodules of a module. This relation was subsequently used to define the concepts of \(G^{*}\)-lifting and \(G^{*}\)-supplemented modules. The same equivalence relation was later used by \textit{A. T. Guroch} and \textit{E. T. Meric} [``Principally Goldie*-lifting modules'', Preprint, \url{arXiv:1405.3819}] to study principally \(G^{*}\)-lifting modules. In the paper under review, an equivalence relation \(\beta_\delta^*\) is defined using the concept of \(\delta\)-smallness, thereby generalizing the relation \(\beta^*\). A module \(M\) is called principally \(\mathcal{G}^{*}\)-\(\delta\)-lifting if for every cyclic submodule \(mR\) of \(M\), there exist a direct summand \(D\) of \(M\) such that \(mR\) is \(\beta_\delta^*\)-equivalent to \(D\). It is shown that every (principally) \(G^{*}\)-lifting module is principally \(\mathcal{G}^{*}\)-\(\delta\)-lifting, but the converse is not true in general. Principally \(\delta\)-lifting modules (as introduced by \textit{H. Inankil} et al. [Vietnam J. Math. 38, 1890201 (2010; Zbl. 1227.16007)]) are shown to be principally \(\mathcal{G}^{*}\)-\(\delta\)-lifting, but the converse does not hold. Conditions for a principally \(\mathcal{G}^{*}\)-\(\delta\)-lifting module to be principally \(\delta\)-lifting are found as well as a relation between \(\mathcal{G}^{*}\)-\(\delta\)-lifting modules and principally \(\delta\)-supplemented modules. Direct summands and homomorphic images of \(\mathcal{G}^{*}\)-\(\delta\)-lifting modules are also investigated. A module is defined to be principally \(\mathcal{G}^{*}\)-\(\delta\)-supplemented if each cyclic submodule \(mR\) of \(M\) has a \(\delta\)-supplement \(N\) in \(M\) such that \(mR\) is \(\beta_\delta^*\)-equivalent to \(N\). The class of principally \(\mathcal{G}^{*}\)-\(\delta\)-lifting modules is shown to be between the class of principally \(\delta\)-lifting modules and the class of principally \(\mathcal{G}^{*}\)-\(\delta\)-supplemented modules, while the last two classes coincide if \(M\) is a \(\pi\)-projective module. Relationships between principally \(\mathcal{G}^{*}\)-\(\delta\)-lifting modules and principally \(\delta\)-semi-perfect rings as well as principally semisimple modules are also found.