Graded Bass Rings And Duality

The Graded ring theory occupies the central part on the study of ring theory. In recent yearsmany algebralists such as J .E.Bjé°Žh, F.Van.Oystaeyen,C.Nastasescu.S.Montgomery. D.Passman. M.Beattie, Liu Shaoxue based the study of graded ring structure on Algebra,which makes great further achivement in this branch. On introducing graded Flamsher module(i.e.graded Bass module) and graded Semiartinian module, the thesis discusses the char- acterization of the according module class. With the help of graded Jocobson radical(Socle), graded maxinial(minimal) submodule, graded cogenerator and graded injective hull, graded Bass(Semiartin) ring is thoroughly characterized. Again using Smash productsExcellent Ex- tensions ect. the author discusses the relations among graded ring R and the rings(R. R~, R#G) which are derived by it. And obtains R. R6. R#G and graded ring I? are identical on the prop- erties about Bass(Semiartin) rings under the conditions of finite grotip G-type strongly graded rings( C is inverse of R,e is the identity of G).Finally. the ideas of C.Menini and A.Del Rio are cited to conclude: If H = ?R5(S = ~ Sj is a G(F)-type graded ring,Q ~ ~Q0 is a gEG eel? gEG eel? higraded R-S-bimodule and nQs define a graded Morita duality between R and S. then R is a graded left Bass ring if and only if S is a graded right Semiartin ring. The thesis is divided into 4 sections.In Section 1. some preliminaries and notations are mentioned.In order to build a good foundation for later sections, it is necessary to cover: the preliminaries of graded ring;Smash products;Excellent Extensions;the backround of graded Bass(Semiartin) rings;graded Morita duality ect. In Section 2 and 3, graded Bass(Semiartin) rings are introduced and characterized. In Section 4, we conclude that under certain coditions, graded left Bass rings are dual to graded right Semiartin rings.