Hereditary Torsion Theory And Semistar-operations On Communitative Ring

In this paper, we studyτ-torsionfree module of hereditary torsion theory byutilizing semistar operations .This thesis into three chapters. In chapter 1, weapply the notion of hereditary torsion theory and Manis valuation ring and theirscorresponding results. In chapter 2, we firstly introduce two notions:τ-moduleandτ-envelope, and discuss some properties of them. Using semistar operationsmethods, we prove that Misτ-torsionfree, Mτisτ-envelope of M, if and only ifMτ= {x∈E(M)|J∈F(L ),Jx∈M}. Secondly, we introduce quasi-τ-fintetype, quasi-τ-Noether module and quasi-τ-Noether ring byτ-envelope of notion,and discuss some basic propertises of these. In chapter 3, we do our work aroundquasi-τ-ideals.Firstly, we prove each quasi-τ-maximal-ideal is a prime ideals; Misτ-torsion module if and only if each quasi-τ-maximal-ideal m, Mm = 0. Fur-thermore, we discuss the ascending chain condition of the quasi-τ-ideal of R, andprove R is quasi-τ-Noether ring if and only if R has ascending chain condition ofthe quasi-τ-ideal. Moreover we introduce the notion ofτ-invertible and PτMRring. Then we obtain the following equivalent conditions: (1) R is PτMR ring;(2) Every nonzero finitely generated regular ideals of R isτ-invertible; (3) Ev-ery nonzero quasi-τ-finite type regular ideals isτ-invertible; (4) R[m] is a Manisvaluation ring for all regular quasi-τ-maximal-ideal m of R; (5) R[m] is a pseudo-valuation ring for all regular quasi-τ-maximal-ideal m of R.