Graded Extensions In KZ^（n）

As an important class of rings,skew group rings are of great significance in the study of non-commutative valuation rings,graded algebra and extensions in graded rings.The question of graded extensions in tensor product was put forward by H.H.Brungs,H.Marubayashi,E.Osmanagic,and they prove that there is a one to one correspondence between Gauss extensions and graded ex-tensions.So it is a new way to study graded extensions in order to study Gauss extensions.Skew Laurent polynomial ring is an important class of skew group ring.In recent years,the research on graded extensions in skew Laurent polynomial rings has made great progress.Graded extensions in skew Laurent polynomial ring K[Z,σ]＝K[X,X-1,σ]were discussed in detail by Guangming Xie and H.Marubagashi etc.They classify A by the following eight different types based on the properties of A1 and A-1:type（a）,type（b）,type（c）,type（d）,type（e）,type（f）,type（g）,type（h）,and describe the structures of each type in detail.The research on graded extensions in K[x₁,x₂;x₁^-1,x₂^-1],graded extensions in K[x₁,x₁^-1]and K[x₂,x₂^-1]are given,and then they discuss their expansion separately.In this paper we will discuss graded extensions in KZ^（n）= K[x₁,…,x_n:x₁^-1,…,x_n-1].If we take the same way in discuss graded extensions in Kn[x₁,x₂;x₁^-1,x₂^-1,when n is large enough,the classification is complicated.The proof is also more difficult.Let K is a field and σ = 1.As the classification in K[Z,σ]by Guangming Xie etc,we classify A in KZ^（n）by the following four different types:type（a）,type（d）,type（e）and generalized type（h）.Then we discuss the properties of graded extensions on these types,the sufficient conditions for existence.We prove that A =（?）u∈Z^（n）H,AuXu is a graded extension of V in KZ^（n）if and only if A is of type（a）,type（d）,type（e）or generalized type（h）graded extension of V in KZ^（n）At last we will give some specific examples of graded extensions of each type in KZ^（n）.This paper is composed of four parts.The first part is the introduction,the second to fourth 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 and the main results of this paper are introduced.Chapter 1 of this paper is composed of two parts.some basic concepts and common lemmas are introduced in the first part,In the second part we will mainly discuss graded extensions of V in KZ（2）.Graded extensions in KZ（2）will be divided into type（a）,type（d）,type（e）and generalized type（h）.The main result is Theorem 1.1:A=（?）u∈Z（2）AuXτ is a graded extension of Vin KZ（2）if and only if A is of type（a）,type（d）,type（e）or generalized type（h）graded extension of V in KZ（2）.In Chapter 2,we will mainly discuss graded extensions of V in KZ^（n）.Similarly,the graded extensions in KZ^（n）will be divided into type（a）,type（d）,type（e）and generalized type（h）.The main result is Theorem 2.1:A=（?）u∈Z^（n）AuXu is a graded extension of Vin KZ^（n）if and only if A is of type（a）,type（d）,type（e）or generalized type（h）graded extension of V in KZ^（n）.In Chapter 3,we will give some specific examples of graded extensions of each type in KZ^（n）.The last part is the concluding remarks.The main work of this paper will be summarized.Also some problems will be put forward.