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On generalized CS-modules. (English) Zbl 1363.16015

A module \(M\) (over a ring \(R\)) is called a CS-module (or an extending module) if every submodule of \(M\) is essential in a direct summand of \(M\). A submodule \(N\) of a module \(M\) is said to be \(\mathcal{G}\)-closed if \(M/N\) is a nonsingular \(R\)-module. A module \(M\) is called a generalized CS-module (or briefly, GCS-module) if any \(\mathcal{G}\)-closed submodule of \(M\) is a direct summand of \(M\). GCS-modules provide a generalization of both CS-modules and singular modules. In this paper, the author considers the question of when a finite direct sum of GCS-modules is again a GCS-module, and obtains various partial results. It is proved that, for a right nonsingular ring \(R\), every right \(R\)-module is a GCS-module if and only if every nonsingular right \(R\)-module is projective. Also, every finitely generated right \(R\)-module is GCS if and only if every finitely generated nonsingular right \(R\)-module is projective, and if moreover the identity of \(R\) is a sum of orthogonal primitive idempotents, then these conditions are equivalent to \(R\) being left and right CS and left and right semi-hereditary.

MSC:

16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16D10 General module theory in associative algebras
16S99 Associative rings and algebras arising under various constructions
16D20 Bimodules in associative algebras
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References:

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