In this presentation, we talk about the proper class of short exact sequences determined by τ-closed submodules in the category of left R-modules. A submodule N of a module M is said to be τ-closed (or τ-pure) provided that M/N is τ-torsion-free, where τ= (T_τ ,F_τ) is a torsion theory. Denote by τ - Closed the class of all short exact sequences 0→A→┴f B→C→0 such that Im f is τ-closed in B. The class τ -Closed of τ-closed short exact sequences need not be a proper class, and hence we consider the smallest proper class P_τ containing τ-Closed, that is, the intersection of all proper classes containing it. We describe the class P_τ in terms of τ-closed submodules: we show that A is a extended τ-closed submodule of B if and only if there is a submodule S of B such that A∩S=0 and B/(A ⊕ S) is τ-torsion-free. Denote by P_τ the class of all short exact sequences 0→A→┴f B→C→0 such that Im f is extended τ-closed in B. It is know that the class P_τ forms a proper class . We prove that P_τ is the smallest proper class containing τ-Closed. Later we show that the smallest proper class P_τ is the proper classes projectively generated by the class of τ-torsion modules and coprojectively generated by the class of τ-torsion-free modules. The structure of ring over which every τ-closed submodule is closed is also characterized.
Anahtar Kelimeler: τ-closed Submodule, Proper Class, Pure Submodule, Goldie’s Torsion Theory, Dickson’s Torsion Theory