CO-SEMISIMPLE MODULES AND GENERALIZED INJECTIVITY

Abstract

Let $\cal F$ be a left Gabriel topology on a ring $R$ and $\cal X$ be a special class of left $R$-modules (for example, the class of all quasi-continuous left $R$-modules in $\sigma[M]$, etc.). Suppose that all left $R$-modules in $\cal X$ are $\cal F$-injective. Then, it is proved in this paper that a left $R$-module $M$ is $\cal F$-co-semisimple (that is, every $\cal F$-cocritical left $R$-module $C$ in $\sigma[M]$ is dense in its $M$-injective hull) if and only if every $\cal F$-torsionfree $\cal F$-finitely cogenerated left $R$-module $N$ in $\sigma[M]$ is dense in its some essential extensions which are in $\cal X$. As a corollary we show that a left $R$-module $M$ is co-semisimple if and only if every finitely cogenerated left $R$-module in $\sigma [M]$ is continuous (or quasi-continuous, or direct-injective, etc.)