τclosed submodule were introduced and studied as generalization of closed submodule, where τ stands for an idempotent radical ([1]). τ – Closed is the class of all short exact sequences 0→A→┴f B→C→0 such that Im f is τclosed in B and P_τ is the smallest proper class containing τClosed. In this presentation, we talk about the relations between the proper class P_τ and some of wellknown proper classes, such as Closed, Pure, Neat. It is known that closed submodules are neat, and neat submodules are closed over Crings ([2]). Moreover, over commutative ring R, closed, neat, coneat and spure submodules are the same if and only if R is noetherian distributive ([4]). A problem related with τclosed submodule is that it is not known when closed submodules are τclosed and τclosed submodules are closed. So, for a hereditary torsion theory τ, we prove the following;
(1) Every τclosed submodule is closed if and only if every singular module is τtorsion 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 τtorsion module is semisimple, (4). if R is a commutative Noetherian ring, then every τclosed submodule is pure if and only if R≅A×B, wherein A is τtorsion ring and B is hereditary C_τ ring. In particular, over commutative Noetherian ring R, every Sclosed 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
