In this presentation, we talk about the proper class of short exact sequences determined by τclosed submodules in the category of left Rmodules. A submodule N of a module M is said to be τclosed (or τpure) provided that M/N is τtorsionfree, 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 τclosed in B. The class τ Closed of τclosed short exact sequences need not be a proper class, and hence we consider the smallest proper class P_τ containing τClosed, that is, the intersection of all proper classes containing it. We describe the class P_τ in terms of τclosed submodules: we show that A is a extended τclosed submodule of B if and only if there is a submodule S of B such that A∩S=0 and B/(A ⊕ S) is τtorsionfree. Denote by P_τ the class of all short exact sequences 0→A→┴f B→C→0 such that Im f is extended τclosed in B. It is know that the class P_τ forms a proper class [4]. We prove that P_τ is the smallest proper class containing τClosed. Later we show that the smallest proper class P_τ is the proper classes projectively generated by the class of τtorsion modules and coprojectively generated by the class of τtorsionfree modules. The structure of ring over which every τclosed submodule is closed is also characterized.
Anahtar Kelimeler: τclosed Submodule, Proper Class, Pure Submodule, Goldie’s Torsion Theory, Dickson’s Torsion Theory
