A Classification Of Graded Extensions In K[z^?2?;?]

As an important class of rings,non-commutative valuation rings are of great significance in the research of non-commutative ring theory.Ring extension is an important part of ring theory.In recent years,the question of non-commutative valuation ring extensions was put forward by H.H.Brungs,G.Torner and M.Schroder.After that,much important progress has been obtained.Grad-ed extensions and Gauss extensions are two kinds of important valuation ring extensions,and there is a one to one correspondence between graded extensions and Gauss extensions.So it suffices to study graded extensions in order to study Gauss extensions.Graded extensions as a kind of special graded algebra has an important research significance.Let Z be the additive group of integers,Aut(K)be the group of automorphisms of division ring K,and ? be a group homomorphism from Z2 to Aut(K),K[Z(2);?]be the skew group ring.Skew Laurent polynomial ring is an important class of ring.The graded extensions of skew Laurent polynomial ring K[x,x-1;?]and the trivial graded extensions of skew group ring K[Z2;?]was discussed in detail by Guangming Xie etc..But for and in the study of the extensions of skew group ring K[Z2;?],only the simplest trivial graded extensions were discussed.The general graded extensions were not.We will describe the structures of each type in detail.In this paper,definitions of each type graded extensions in K[Z(2);?]are given.We will also describe the concrete structures of graded extensions in K[Z(2);?]in detail.At last,we will give some examples of each type.This paper is composed of five parts.The first part is introduction.The second,third and fourth part are the main body of this paper.And the last part is the concluding remarks.In introduction,some of the research background,research significance of this paper are introduced.In Chapter 1,some of the fundamental lemmas,definitions,properties of graded extensions are introduced.In Chapter 2,we will study the properties of each kind of graded extensions in K[Z(2);?],and describe the structures of each type in detail.In Chapter 3,some specific examples of graded extensions of each type in K[Z(2);?]will be given.The last part is the concluding remarks.The main work of this paper will be summarized.Also some problems will be put forward.