Graded Extensions In K[χ₁,χ₂;χ₁^-1,χ₂^-1]

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. At the end of last century, 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. Graded 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. Skew Laurent polynomial ring is an important class of ring. Graded extensions in skew Laurent polynomial ring K[x,x-1;σ] and trivial graded extensions in skew group ring K[Z2; a] were discussed in detail by Guangming Xie etc., where Z is the additive group of integers, Aut(K) is the group of automorphisms of division ring K, and σ is a group homomorphism from Z2 to Aut(K). But for extensions in skew group ring K[Z2; σ], however, only the simplest trivial graded extensions were discussed. General graded extensions were not. We will give a complete classification of graded extensions in K[x1,x2;x1-1,Xx-1], and describe the structures of each type in detail.In this thesis, graded extensions in K[x1,x1-1) and K[x2,x2-1] are given. We will discuss the existence and uniqueness of graded extensions in K[x1, x2; x1-1, x2-1]. We will also describe the concrete structures of graded extensions in K[x1, x2;x1-1, x2-1]. At last, we will give some examples of each type.This thesis is composed of four parts. The first part is introduction. The second and third part are the main bodies of this thesis. And the last part is the concluding remarks.In Chapter 1, some of the research background, research significance and main results are introduced.In Chapter 2, we will give a complete classification of graded extensions in K[x1,x2; x1-1,x2-1],and describe the structures of each type in detail. The results in this section were published in Journal of Guangxi Normal University(2015,33(1):74-79).In Chapter 3, some specific examples of graded extensions of each type in K[x1, x1; x1-1,x2-will be given.The last part is the concluding remarks. The main work of this paper will be summa-rized. Also some problems will be put forward.