Abstract
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 τ-extending module, where τ is a hereditary torsion theory on Mod-R. An R-module M is called type 2 τ-extending if every type 2 τ-closed submodule of M is a direct summand of M. If τI is the torsion theory on Mod-R corresponding to an idempotent ideal I of R and M is a type 2 τI-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 τG we give an example of a module that is type 2 τG-extending but not extending.
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