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torsion theory; extending module; closed submodule
An $R$-module $M$ is said to be an extending module if every closed submodule of $M$ is a direct summand. In this paper we introduce and investigate the concept of a type 2 $\tau $-extending module, where $\tau $ is a hereditary torsion theory on $\mathop {\text{Mod}}$-$R$. An $R$-module $M$ is called type 2 $\tau $-extending if every type 2 $\tau $-closed submodule of $M$ is a direct summand of $M$. If $\tau _I$ is the torsion theory on $\mathop {\text{Mod}}$-$R$ corresponding to an idempotent ideal $I$ of $R$ and $M$ is a type 2 $\tau _I$-extending $R$-module, then the question of whether or not $M/MI$ is an extending $R/I$-module is investigated. In particular, for the Goldie torsion theory $\tau _G$ we give an example of a module that is type 2 ${\tau }_G$-extending but not extending.
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