Gauss Extensions In K(（?）^（2）,σ)

The theory of ring extensions is an important part of ring theory.Assume that a crossed product K*G of G over K has a right quotient skew field Q(K*G).H.H.Brungs,H.Marubayshi,E.Osmanagic defined the concept of graded extension on K*G,and proved that there was a one-to-one correspondence between the set of graded extensions and the set of Gauss extensions.It provided an important method to study Gauss extensions.Skew Laurent polynomial ring is an important ring.According to the properties of A1 and A-1,G.Xie and H.Marubayshi classified the graded extensions on K[X,X-1;σ]into types(a),(b),(c),(d),(e),(f),(g),(h),eight kinds of graded extensions.The structures of graded extensions of each type were also studied in detail.In this paper,we will study Gauss extensions in K(Z(2),σ).Let V be a total valuation ring of a division ring K,σ:Z(2)→Aut(K)be a group homomorphism,and K[Z(2),σ]be the skew group ring of Z(2)over K.We can study the Gauss extensions in K(Z(2),σ)by studying the graded extensions in K[X(1,0),X(-1,0);σ(1,0)]and the graded extensions in K(X(0,1),X(0;-1);σ(0,1)].This paper is divided into five parts.The first part is the introduction.The second,the third and fourth parts are the main part.The last part is the conclusion and prospect.In the first chapter,we mainly introduce research background,the basic concepts and lemmas involved in this paper.In Chapter 2,first of all,we will prove that K[Z(2),σ]has a quotient ring K(Z(2),σ).Then we will discuss Gauss extensions in K(Z(2),σ).Let τ=σ(1,0),θ=σ(0,1),X=X(1,0),Y=X(0,1).So we can view D=K(X,τ)as a division subring of K(Z(2),σ),and K(Z(2),σ)=D(Y,θ).Let R be a Gauss extension of V in K(ZZ(2),σ),R∩K[Z(2),σ]=A=⊕(i,j)∈Z(2)A(i,j)XiYj,and let R ∩ DYj=SjYj.We will prove that B=⊕j∈z SjYj is a graded extension of S0 in D[Y,Y-1;θ].On the other hand,the converse is also true.If let E=K(Y,θ),we will also have a similar discussion.In Chapter 3,let B1,B2 be one of graded extensions of types(a),(b),(c),(d),(e),(g),or(h)and the other one be another extension.We will discuss the compatibility of B1 and B2 under some special conditions.And we will discuss the relationship of B1,B2 with the corresponding graded extension A.In Chapter 4,we will give some examples with B1,B2 compatible or B1,B2 not compatible.In the last part,we will summarize the main results of this paper,and put forward some questions that can be further studied.