A lemma on indecomposable reflexive modules and applications. (Q1885180)

A lemma is proved which shows when an indecomposable left \(S\)-module \(_SM\) is \(U\)-reflexive, where \({_SU_R}={_SV_R}\oplus{_SW_R}\) (in terms of \(V\)-and \(W\)-reflexivity). As applications, diverse bimodules admitting few reflexive modules are constructed. All results (a series of examples) are related with cotilting, quasi-cotilting and locally quasi-cotilting bimodules. For example, a locally quasi-cotilting bimodule \(_SU_R\) with the following properties is constructed: (i) \(\text{Cogen\,}U_R\) is a torsionfree class; (ii) \(\text{Cogen\,}{_SU}\) is not a torsionfree class; (iii) \(0\neq\text{Im}(\text{Ext}^1_S(-,U)\circ\Hom_R(-,U))\subseteq \text{Cogen\,}U_R\) and \(0\neq\text{Im}(\text{Ext}^1_R(-,U)\circ\Hom_S(-,U))\subseteq \text{Ker}(\text{Hom}_S(-,U))\). A special technique permits to obtain quite different situations related with reflexive modules.