Some Discussion About Selforthogonal Modules

Auslander and Reiten put forward the concept of κ-torsionfree modules when they studied the stable modules in the later 1960's. It is a generalization of torsionless modules and reflexive modules. A lot of articles have studied the property of κ-torsionfree modules, especially, the relation between the extension closure of T_ω~κ(A) and grade and strong grade. In ref[l], the author defmites the notion of fc-torsionfree modules relative to the faithfully balanced selforthogonal bimodule (?)ωΓ. Similarly, we are interested in investigating the property of ω-κ-torsionfree modules and the relation between the extension closure of T_ω~κ(A) and grade and s.grade relative to (?)ωΓ.The notion of syzygy modules is very important in homological algebra. It is very useful when we study the complex and the functors of Ext and Tor. Since the notion of syzygy modules relative to (?)ωΓ had been put forward, the theory of syzygy modules was deeply developed. In fact, we have proved T~κ(A) (?) Ω~κ(?) and T_ω~κ(?) (?) Ω_ω~κ(?), and many mathemetical workers study them together for they are related tightily. Professor Huang Zhaoyong do a lot of works in this field, and my interesting in studing in this field are stimuluted by Prof. Huang's works just. We will divided the thesis in four sections.In the first section, we will introduce oue main opinion and outline main results in this thesis.In the second section, we mainly study the selforthogonal modules with finite injective dimension, especially, dicuss some questions no cotilting modules. In ref[l], the author gains a equivalent condition that a finitely generated module is a ω-torsionless module or ω-reflexive module. We will generalize this theorem, and we will get a equivalent condition that a finitely generated module is a ω-κ-torsionfree module. Furthermore, we also dicuss some questions no the cotilting bimodule (?)ωΓ when l.id(ω) ≤κ. And the...