Pure Cones Of Q(I) And The Corresponding Trivial Graded Extensions

The theory of graded rings of groups is the convergence of group theory and ring theory. The results of graded rings are influential to the research of group theory and ring theory. Graded extensions and Gauss extensions are two kinds of importent extensions of rings. The research of pure cones is importent for us to describe the two extensions above. Let Q be the addtitive group of rational numbers, K be a division ring, Aut(K) be the group of automorphisioms of K,σbe a group homomorphism from G to Aut(K), K[Q(n),σ] be the skew group ring of G over K.In this paper we mainly discuss the pure cones of Q(n) and the trivial graded extensions of K[Q(n),σ]. Furthermore, we study the properties of pure cones of Q(I) and investigate pure cones of R and the corresponding graded extensions.This paper is composed of six parts. The first part is introduction. Second to fifth parts are the body of this paper. And the last part is the concluding remarks.In introduction, some of the research background, research significance and main re-search results of this paper are introduced.In Chapter 1, we introduce the definition of trivial graded extension, discuss the rela-tionship of pure cones and trivial graded extensions. The main results are the following:Theorem 1.4 Let G be a group with a pure cone, K[G,σ] has a left quotient ring K(G,σ), V≠K is a total valuation ring of K. Then there is a one-to-one correspondence between the set of all trivial graded extensions of V in K[G,σ] and the set of all pure cones in G.In Chapter 2, we descripte the pure cones of Q and Q(2) and give the corresponding trivial graded extensions of Q and Q(2). The main results are the following:Theorem 2.1 Let P be a subset of Q. Then P is a pure cone of Q if and only if P= Q0+ or P= Q0-, where Q0+={x≥0| x∈Q}, Q0--={x≤0| x∈Q}.Theorem 2.2 P is a pure cone of Q if and only if there exists a non-zero number a∈R such that P={x∈Q| ax> 0}∪{0}. Theorem 2.6 Let P be a subset of Q(2),I1={(0,1),(0,-1)},I2={(1,0),(-1,0)}. Then P is a pure cone of Q(2) if and only if there exist r∈S,α∈I1∩P,β∈I2∩P such that P=Pr or P=Pr' where S=[0,∞)∪{∞},Theorem 2.7 P2 is a pure cone of Q(2) if and only if there exist a,b∈R which are not all zero,such that P2={(x,y)∈Q(2)|ax+by>0}∪{(x,y)∈Q(2)|ax+by=0,if b=0, then y≥0,else x≥0} or P2={(x,y)∈Q(2)|ax+by>0}∪{(x,y)∈Q(2)|ax+by=0, if b=0,then y≤0,else x≤0}.Theorem 2.8 P2 is a pure cone of Q(2) if and only if there exist a,b∈R which are not all zero,such that P2={(x,y)∈Q(2)|ax+by>0}∪Pax+by=0,where Pax+by=0 is a pure cone of group {(x,y)∈Q(2)|ax+by=0}.Theorem 2.10 Let V≠K be a total valuation ring of K,let subset of K[Q,σ] and A0=V.Then A is a trivial graded extension of V over if and only ifTheorem 2.11 Let V≠K be a total valuation ring of K,and suppose that K[Q(2),σ] has a left quotient K(Q(2),σ).Let A= be a subset of K[Q(2),σ]and A(0,0)=V.Then A is a trivial graded extension of V over K[Q(2),σ]if and only if there exists r∈[0,∞)∪{∞},α∈I1,β∈I2 such that one of the following properties holds:In Chapter 3,we give a complete descriptions of the pure cones of Q(n) and the corre-sponding trivial graded extensions.The main results are the following:Theorem 3.7 Pn is a pure cone of Q(n) if and only if there exsit which are not all zero,such that where is a pure cone of the group {(x1,x2,…,xn)Theorem 3.8 Let V≠K be a total valuation ring of K and suppose that K[Q(n),σ] has a left quotient K(Q(n),σ).Let be a subset of K[Q(n),σ]and A(0,0,…,0)=V.Then A is a trivial graded extension of V if and only if there exsits a pure In Chapter 4, we give some properties of pure cones of Q(I), especially, discuss pure cones of R. The main results are the following:Property 4.1 P is a pure cone of Q(I) if and only if for anyα,β∈Q(I), (Qα+Qβ)∩P is a pure cone of Qα+Qβ.Property 4.2 Let I be an index set and let (xα)α∈I be a basis of Q(I) of vector space of Q. Then P is a pure cone of Q(I) if and only if for any (xs)s∈F (F(?) I,F is a finite set), is a pure cone ofProperty 4.4 Let I be an index set which is infinite, let J={P|P is a pure cone of Q(I)}. Then Card J=2CardI= Card(2I).Property 4.5 Let V≠K be a total valuation ring of K and suppose that K[Q(I),σ] has a left quotient K(Q(I),σ). Let (?)={B|B is a trivial graded extension of K [Q(I),σ]}. Then Card (?)=2CardI=Card(2I).Corollary 4.6 Let M={P|P is a pure cone of R}. Then Card M= 2N.Corollary 4.7 Let V≠K be a total valuation ring of K and suppose that K[R,σ] has a left quotient K(R,σ). Let (?)={E| E is a trivial graded extension of K[R,σ]}. Then Card (?)=2N.The last part is the concluding remarks, the main work of this paper is summarized, and the plan to the next work is roughly envisaged.