τ-closed submodule were introduced and studied as generalization of closed submodule, where τ stands for an idempotent radical (). τ – Closed is the class of all short exact sequences 0→A→┴f B→C→0 such that Im f is τ-closed in B and P_τ is the smallest proper class containing τ-Closed. In this presentation, we talk about the relations between the proper class P_τ and some of well-known proper classes, such as Closed, Pure, Neat. It is known that closed submodules are neat, and neat submodules are closed over C-rings (). Moreover, over commutative ring R, closed, neat, coneat and s-pure submodules are the same if and only if R is noetherian distributive (). A problem related with τ-closed submodule is that it is not known when closed submodules are τ-closed and τ-closed submodules are closed. So, for a hereditary torsion theory τ, we prove the following;
(1) Every τ-closed submodule is closed if and only if every singular module is τ-torsion module, (2) If τ(R_R )=0, then P_σ = P_τ if and only if R is a right C_τ, (3) if τ(R_R )=0, then P_τ=Neat if and only if each τ-torsion module is semisimple, (4). if R is a commutative Noetherian ring, then every τ-closed submodule is pure if and only if R≅A×B, wherein A is τ-torsion ring and B is hereditary C_τ ring. In particular, over commutative Noetherian ring R, every S-closed submodule is pure if and only if R≅A×B, wherein A is Goldie torsion ring and B is hereditary ring.
Anahtar Kelimeler: (τ-) closed Submodule, Proper Class, Pure Submodule, Goldie’s Torsion Theory, Dickson’s Torsion Theory