## Taiwanese Journal of Mathematics

### 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.)

#### Article information

Source
Taiwanese J. Math., Volume 3, Number 3 (1999), 357-366.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500407134

Digital Object Identifier
doi:10.11650/twjm/1500407134

Mathematical Reviews number (MathSciNet)
MR1706041

Zentralblatt MATH identifier
0937.16006

#### Citation

Liu, Zhongkui; Ahsan, Javed. CO-SEMISIMPLE MODULES AND GENERALIZED INJECTIVITY. Taiwanese J. Math. 3 (1999), no. 3, 357--366. doi:10.11650/twjm/1500407134. https://projecteuclid.org/euclid.twjm/1500407134