Hereditary Pretorsion Classes And Hereditary Torsion Classes

R.Colpi gave some conditions that Gen(R^P) is closed under submodules and extension when R^P is a *-module. We gave the coditions that Gen(R^P) is closed under submodules and extension to be a hereditary torsion class, when R^P is an arbitrary module. And we studied some properties of it. This paper is divided into three chapters.In the first chapter, wo introduced the background of this paper and some definitions.In the second chapter, we discussed the necessary and sufficient conditions for that Gen(R^P) is a hereditary (pre)torsion class. And then gave some properties of it.The main results as following:Theorem 2.1.4 Let P be a left R-module and lot S=End(R^P), then the following conditions are equivalent:(1)Gen(R^P) is a hereditary pretorsion class.(2)P_S is flat and S-Mittag-Leffler, Gen(R^P)=Pres(R^P), and TH preserves cpimorphismin Gen(R^P).(3)P_S is fiat and Gen(R^P)=Stat(P)Theorem 2.2.1 Let. P be a left R-module and let S=End(R^P), if Gen(R^P) is a hereditary pretorsion class, then Gen(R^P) is a hereditary torsion class if only if Gen(R^P)(?)Ker P(?)Ext_R¹(P,-).Theorem 2.2.2 Let P be a left R-module and let S=End(R^P), then Gen(R^P) is a hereditary torsion class if and only if the following coditions are satisfied:(1)P_S is flat and S-Mittag-Leffler;(2)Gen(R^P)=Pres(R^P)(?)Ker P(?)_SExt_R¹(P, -).The acording torsion-free class is Ker Hom_R(P, -) when those conditions above arc satisfied.Theorem 2.2.10 Let R be a ring and let P be a module of finite length over R, the Gen(R^P) is a hereditary torsion class if and only if there is a quasi-progenerators R^T satisfying Gcn(R^P)=Gen(R^T)(?)Ker ExtR_<sup>1(T,- )Theorem 2.3.5 Let R be a Artin ring and let P be a left R-module, S=End(R^P), then Gen(R^P) is a hereditary torsion class if and only if the following coditions are satisfied:(1)Ps is a finitely generated projective module;(2)Gen(R^P)=Pres(R^P)(?)Ker P(?)_SExt_R¹(P,-).Theorem 2.3.10 Let R be a Artin ring and let P bo a finitely generated moduleover R. If Gen(R^P) is a hereditary torsion class, then P₀-res.dim(M)=pd_SH_p0M for all M∈Gen(R^P) (P₀is the quasi-progenerator in Theorem 2.2.10).Corollary 2.3.11 let P be a finitely generated module over a Artin ring R, and Gen(R^P) is a hereditary torsion class. If P₀-res.dim(M)≤n for all M∈Gen(R^P) (Pois the quasi-progenerator in Theorem 2.2.10). then gdS≤n (S=End(R^P₀)).In the third chapter, we inverstigated some other hereditary (pre)torsion classes over the endomorphism and the biendomorphism ring of R^P. The main results as following:Corollary 3.1.2 Let P be a left R-module and let S=End(R^P), if Gen(R^P) is a hereditary pretorsion class, then:(1)(Ker P(?)_S-,Cogen(S^P*)) is a torsion class cogenerated by Copres(S^P*);(2)(Ker P(?)_S-,Cogen(S^P*)) is a hereditary torsion class.Corollary 3.1.2 If Gen(R^P) is a hereditary pretorsion class in R-Mod, then Gen(R^P) is a hereditary pretorsion class in B-Mod (B=BiEnd(R^P)).