Pure Cones And Trivial Graded Extensions

Graded extensions and Gauss extensions are two important extensions of rings.The study of pure cones is significant for characterizing graded extensions and gauss extensions.Let V be a total valuation ring of a skew field K,V?K,and let G be a group,?:Q(n)? Aut(K)be a group homomorphism,where Aut(K)is the automorphism group of K.Suppose that the skew group ring K[G,?]of G on K has a left quotient skew field Q(K[G,?]).In this paper,we research on cases of G=Q(n)and G=Z(n).Firstly,we give a complete description of pure cones of Q(n).Based on this,then we characterize the graded extensions in K[Q(n),?].Finally,we study the pure cones of Z(n)and the trivial graded extensions in K[Z(n)?].This paper is composed of six chapters.The first chapter is the introduction,the second chapter to fifth chapter are the process of our study,the last chapter is the conclusion.In chapter 1,we introduce the research background and significance of this paper.In chapter 2,we introduce the definitions of cones,pure cones,total valuation rings,skew group rings,left quotient skew fields,graded extensions,trivial graded extensions,graded subrings,etc.And we study the pure cones of Q(n).The main results are as follows.Theorem 2.1 Let P be a pure cone of Q(n).Then there are real numbers a1,a2,…,an that are not all 0,such that(1)(?)u=(u1,u2,…,un)? Q(n),if a1u1+a2u2+…anun>0,then u?P.(2)(?)u=(u1,u2,…,un)? Q(n),if a1u1+a2u2+?…anun<0,then u(?)P.(3)Suppose S={(u1,u2},…,un)|a1u1+a2u2+…+anun=0,ui?Q,i=1,2,…,n}.Then P ? S is a pure cone of S.Theorem 2.2 Let a1,a2,…,an be real numbers that are not all 0,P be a subset of Q(n),S={(u1,u2,…,un)|a1u1+a2U2+…+anun=0,ui?Q,i=1,2,…,n}.(1)For any u=(u1,u2,…,un)?Q(n),if a1u1+a2u2+…+anun>0,then set u ? P;(2)For any u=(u1,u2,…,un)?Q(n),if a1u1+a2u2+…+anun<0,then set u(?)P;(3)Suppose that P ? S is a pure cone of S.Then P is a pure cone of Q(n).In chapter 3,we characterize the set of trivial graded extensions in K[(Q(n),?]and the set of all pure cones of Q(n).The main results are as follows.Theorem 3.1 Let V be a total valuation ring of skew field K,V(?)K.Then there is a one-to-one correspondence between the set of trivial graded extensions in K[Q(n),?]and the set of all pure cones of Q(n).Theorem 3.2 Let A=?u?Q(n)AuXu be a trivial graded extension of V in K[Q(n),?].Then there are real numbers a1,a2,…,an that are not all 0,such that(1)For any u=(u1,u2,…,un),a1u1+a2u2+…+anun>0 implies Au=K.(2)For any u=(u1,u2,…,un),a1u1+a2u2+…+anun<0 implies Au=0.(3)Suppose S={(u1,u2,…,un)|a1u1+a2u2+…+anun=0,ui?Q,i=1,2,…,n}.Then B=?u?SAuXu is a trivial graded extension of V in K[S,?].Theorem 3.3 Let a1,a2,…,an be real numbers that are not all 0,S={(u1,u2,…,un)|a1u1+a2u2+…+anun=0,ui?Q,i=1,2,…,n}.(1)For any u=(u1,u2,…,un),if a1u1+a2u2+…+anun>0,then set Au=K;(2)For any u=(u1,u2,…,un),if a1u1+a2u2+…+anun<0,then set Au=0;(3)Suppose that B=(?)u?SAuXu is a trivial graded extension of V in K[S,?].Then A=?u?Q(n)AuXu is a trivial graded extension of V in K[Q(n),?].In chapter 4,we characterize the pure cones of Z(n),the main results are as follows:Theorem 4.1 Suppose that P is a pure cone of Z(n),then there are real numbers a1,a2,…,an that are not all 0,such that(1)(?)u=(u1,u2,…,un)?Z(n),if a1u1+a2u2+…+anun>0,then u ? P.(2)(?)u=(u1,u2,…,un)?Z(n),if a1u1+a2u2+…+anun<0,then u(?)P.(3)Suppose S={(u1,u2,…,un)|a1u1+a2u2+…+anun=0,ui?Z,i=1,2,…,n}.Then P ? S is a pure cone of S.In chapter 5,we characterize the trivial graded extensions in K[Z(n),?],and we get the trivial graded extensions of V in K[Z(n),?].The main results are as follows:Theorem 5.1 Suppose that V is a total valuation ring of a skew field K,V?K,(?)is the set of trivial graded extensions in K[Z(n),?].Then the cardinal number of(?)is(?).Theorem 5.2 Let A=?u?Z(n)AuXu be a trivial graded extensions of V in K[Z(n),?].Then there are real numbers a1,a2,…,an that are not all 0,such that(1)For any u=(u1,u2,…,un),a1u1+a2u2+…+anun>0 implies Au=K.(2)For any u=(u1,u2,…,un),a1u1+a2u2+…+anun<0 implies Au=0.(3)Suppose S={(u1,u2,…,un)|a1u1+a2u2+…+anun=0,ui?Z,i=1,2,…,n}.Then B=?u?SAuXu is a trivial graded extension of V in K[S,?].Theorem 5.3 Let a1,a2,…,an be real numbers that are not all 0,S = {(u1,u2,…,un)|a1u1 + a2u2+…+anun = 0,ui?Z,i = 1,2,…,n}.(1)For any u =(u1,u2,…,un),if a1u1 + a2u2+…+anun>0,then set Au = K;(2)For any u =(u1,u2,…,un),if a1u1+ a2u2+…+ anun<0,then set Au = 0;(3)Suppose that B =?u?SAuXu is a trivial graded extension of V in K[S,?].Then A =?u?Z(n)AuXu is a trivial graded extension of V in K[Z(n),?].