ON W-C-S. MODULES

Abstract

A module is a CS-module (or an extending, or a module with (C 1 )) if every submodule is essential in a direct summand (equivalently if every closed submodule is a direct summand). CS-modules generalize injective, and quasi-injective modules. A module M is continuous if it satisfies (C1) and the following condition (C2 ): if a submodule A of M is isomorphic to a summand of M, then A is a summand of M. A module M is quasi-continouos if it satisfies (C 1) and the following condition (C3): If M1 and M2 are summands of M such that M 1 n M2 = 0, then M 1 EB M 2 is a summand of M. CS­ modules have been studied by many authors, e. g. M.A. Kamal [38], [39], [40], [41], and M. A. Kamal, and B. J. MUller [35], [36], [37], have studied CS-modules over integral domains, and over Noetherian ring. A. W. Chatters, and C. R. Hajamavis [5], have studied CS-Rings. M. Harada and his collaborates have also studied modules with extending properties, which are, in some ways, related to CS-modules. S. H. Mohamed and T. Bouhy [44], have studied continuous modules. S. H. Mohamed and B. J.