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On weak injectivity and weak projectivity. (English) Zbl 0985.16002
Elaydi, S. (ed.) et al., Proceedings of the second Palestinian international conference, West Bank, Palestine, August 19-23, 1998. Singapore: World Scientific. 196-206 (2000).
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 S. K. Jain and 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 S. K. Jain, S. R. López-Permouth and 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.
For the entire collection see [Zbl 0940.00021].
MSC:
 16D50 Injective modules, self-injective associative rings 16D40 Free, projective, and flat modules and ideals in associative algebras 16D90 Module categories in associative algebras 16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras