Abstract
In this work we introduce a new concept, namely, τ s τs -extending modules (rings) which is torsion-theoretic analogues of extending modules and then we extend many results from extending modules to this new concept. For instance we show that for any ring R R with unit, if $_{R}R$ R is purely τ s τs -extending if and only if every cyclic τ τ -nonsingular R R -module is flat. Also, we make a classification for the direct sums of the rings to be purely τ s τs -extending.
Highlights
Injective modules have been intensively studied in the 1960s and 1970s in module theory and more generally in algebra
In this work we introduce a new concept, namely, purely τ s-extending modules which is torsion-theoretic analogues of extending modules and we extend many results from extending modules to this new concept
Clark [8] defined a module M is purely extending if every submodule of M is essential in a pure submodule of M, equivalently every closed submodule of M is pure in M
Summary
Injective modules have been intensively studied in the 1960s and 1970s in module theory and more generally in algebra. As a generalization of injective modules, extending modules (CS), that is every closed submodule is a direct summand, have been studied widely in last three decades. Pure submodule, closed submodule, (non)singular module, extending module, torsion theory. A module M is called purely y-extending if every s-closed submodule of M is pure in M. We use s-closed submodule and purely s-extending module instead of y-closed submodule and purely y-extending module in the sense of Al-Bahrani [1]. Additional and unexplained terminology the reader is referred to [3] or [30] for module and ring theory, [19] and [28] for torsion theory, [15] for extending modules and [26] for homological algebra
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