Graded Extensions In K[Q, 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 H. H. Brungs, G.Torner and G. Schroder, and some great progress have been achieved. Graded extensions and Gauss extensions are two kinds of important ring extensions, and 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. Let Q be the filed of rational numbers, Aut(K) be the group of automorphisms of division ring K,σbe a group homomorphism from Q to Aut(K), and K[Q,σ] be the skew group ring of Q over K. Let K(Q,σ) be the quotient ring of K[Q,σ]. We will discuss the graded extensions of skew group ring K[Q,σ] in this paper.We divide graded extensions in K[Q,σ] into type (Ⅰ) and type (Ⅱ). Graded extensions of type (a), type (b), type (c), type (d), type (e), type (f), type (g), type (h) and type (t) in K[Q,σ] are defined as in [17] and [18]. And we will discuss the relation of this two classifications. Firstly, we will prove that graded extensions of type (Ⅰ) in K[Q,σ] only contain graded extensions of type (a), type (b), type (c), type (d), type (f), type (g) and type (t). Secondly, we will discuss that graded extensions of type (Ⅱ) in K[Q,σ]. This section contains description of the graded maps of Q and corresponding graded extensions of type (e), and description of graded extensions of type (h) and its properties. Finally, we will provide concrete examples of graded extensions of type (a), type (b), type (c), type (d), type (e), type (f), type (g) and type (t).This paper is composed of four parts. 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 Part I, some of the research background, the significance of this paper and the main results of this paper are introduced.In Chapter 1 of this paper, we will mainly discuss the graded extensions of type (Ⅰ) in K[Q,σ]. The main results are the following:Theorems 1.7,1.8,1.9,1.10,1.11,1.12,1.13 and 1.14.These theorems (Theorem 1.7-Theorem 1.13) show that subclasses of graded extensions of type (a), type (b), type (c), type (d), type (f), type (g) and type (t) are still the same to themselves. Theorem 1.14 will show that graded extensions of type (Ⅰ) in K[Q,σ] only contain graded extensions of type (a), type (b), type (c), type (d), type (f), type (g) and type (t).In Chapter 2, we will mainly discuss graded extensions of type (Ⅱ) in K[Q,σ]. This section contains description of graded maps of Q and Corresponding graded extensions of type (e). The main results are the following:Theorems 2.5 and 2.9.Theorem 2.5 shows that any graded map of Q is contained in the set of {fd,fd(1),fd(-1)│d is a real number}.Theorem 2.9 give a sufficient and necessary condition for a graded extension in K[Q,σ] to be a graded extension of type (e). Some results in this section were published in Journal of Guangxi Normal University (2010,28(2):42-46).The last part is concluding remarks. We will give a summary of this research. Also, we will put forward some unsolved questions.