Characterizations of F-V-rings by Quasi-continuous Modules

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Characterizations of F-V-rings by Quasi-continuous Modules

Let R be a ring and F a left gabriel topology on R.According to Garcia Hernandez and gomez Pardo (Comm. Algebra, 13 (1) (1985), 59-83), we say that R is an F-V-ring if every simple object of the category (R, F)-Mod is infective. In this peper,we characterize F-V-rings via quasi-continuity of left R-modules. We show that, if F is a left Gabriel topology on R such that all quasi-continuous left R-modules are F-injective. then R is an F-V-ring if and only if every F-torsionfree and F-finitely cogenerated left R-module is dense in its some essential extensions which are quasi-continuous. As a corollary we show that a ring R is a V-ring if and only if every finitely cogenerated left R-module is continuous, and if and only if every finitely cogenerated left R-module is quasi-continuous.