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Extending modules relative to a torsion theory. (English) Zbl 1166.16014

Summary: 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 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 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.

MSC:

16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16D50 Injective modules, self-injective associative rings
16D90 Module categories in associative algebras
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