Graded Extensions In K(Z⁽²⁾,σ)

As an important class of rings, non-commutative valuation rings are of great significance in the study of non-commutative ring theory. Recently, the question of non-commutative valuation ring extensions was put forward by Brungs, Torner and Schroder, and some great progress have been achieved. Gauss extensions is a type of non-commutative ring extensions with some good properties. The research of Guass extensions is of great significance for studying non-commutative valuation extensions. As we knew, there is a one to one correspondence between Gauss extensions and graded extensions. So it suffices to study graded extensions in order to study Gauss extensions. On the other hand, Graded extensions is a special type of graded algebra. The research of graded extensions itself is of great value. Let Z be the additive group of integers, Aut(K) be the group of automorphisms of division ring K,σbe a group homomorphism from Z(2) to Aut(K), and K[Z(2),σ] be the skew group ring of Z(2) over K. Assume that K[Z(2),σ] has left quotient ring K(Z(2),σ). We will mainly discuss the graded extensions of skew group ring K[Z(2),σ] in this paper. Firstly,we will give a complete description of the pure cones of Z(2); Secondly, we will prove that there is a one to one correspondence between the set of all trivial graded extensions of V in K[Z(2),σ] and the set of all pure cones in Z(2), and give a complete description of the trivial graded extensions of V in K[Z(2),σ]; Finally, we will discuss graded extensions over K[Z(2),σ] under some conditions.This paper is composed of four parts, where the first part is the introduction, the second and third part are the main body of this paper, and the last part is the concluding remarks.In partⅠ, i.e Introduction, some of the research background, the significance of this paper and the main results of the study are introduced.In Chapter 1 of this paper, we will mainly discuss the pure cones of Z(2) and the trivial graded extensions over K[Z(2),σ]. The main results are the following:Theorem 1.2.1,1.3.1,1.3.2 and 1.3.3. Some results in this section were published in Journal of Guangxi Normal University (2009, 27(4):36-40).In Chapter 2, we will discuss graded extensions in K[Z(2),σ] with some specific conditions. Let Vbe a total valuation ring of a division ring K.Setα=(0,1),β=(1,0),σbe a group homomorphism from Z(2)to Aut(K).Let be a graded extension ofV in K[Xα,-Xα,σ(α)], be a graded extension ofV in K[Xβ,X-β,σ(β)]with W,U be overring of V Set be a subset of K[Z(2),σ]with C(0,0)=V.Suppose Ciα=Aiα,Cjβ=Bjβfor any i,j∈Z.When A and B satisfy one of the following four conditions, we discuss the necessary and sufficient condition for C to be a graded extension of V in K[Z(2),σ]:(a)Aiα=Bjβ=V for any i,j∈Z.(b)Aiα=V for any i∈Z；Bjβ=V,B-jβ=J(V)for any j∈N.(c)W(?)V,Aiα,Bjβ=W,A-iα=B-jβ=J(W)for any i,j∈N.(d)Wu(?)U(?)V,Aiα=W,Bjβ=U,A-iα=J(W),B-jβ=J(U)for any i,j∈N.The last part is concluding remarks.We will give a summary of this research.Also,we will put forward some unsolved questions.