On weak injectivity and weak projectivity (Q2710260)

Given a ring \(R\) and a right \(R\)-module \(M\), \(M\) is called weakly semisimple if every module \(X\in\sigma[M]\) is weakly injective in \(\sigma[M]\). Weakly injective modules, a generalization of injective modules, were introduced by \textit{S. K. Jain} and \textit{S. R. López-Permouth} [in J. Algebra 128, No. 1, 257-269 (1990; Zbl 0698.16012)], and their dual concept, weakly projective modules, were introduced and studied by \textit{S. K. Jain}, \textit{S. R. López-Permouth} and \textit{M. H. Saleh} [in Ring theory, Proc. Ohio State-Denison math. conf., World Scientific, 200-208 (1993; Zbl 0853.16004)]. In this paper the authors introduce cotightness as a generalization of weak projectivity and study the basic concepts of weak injectivity (tightness) and weak projectivity (cotightness) in the context of \(\sigma[M]\), the full subcategory in Mod-\(R\) subgenerated by a module \(M\). They also provide several characterizations of semisimple and weakly semisimple modules in terms of tight and cotight modules.NEWLINENEWLINEFor the entire collection see [Zbl 0940.00021].