Cones And Subrings And Over-rings Of Graded Extensions

Let V be a total valuation ring of a skew field of K,G be an additive group with a pure cone,? be a group homomorphism from G to Aut(K).Assume that the skew group ring K[G,?]of G on K has a left quotient skew field Q(K[G,?]).Let A=(?)u?G AuXu be a graded extension of V in K[G,?],P1 be a cone of Q(n).In this paper,the subrings and over-rings of A are studied,and it is shown that there is a one-to-one correspondence between the sets of subrings and over-rings of A and the sets of certain cones of the corresponding groups.In particular,we characterize the cones of Q(n)and the subrings amd over-rings of graded extensions on K[Q(n),?].This paper is mainly divided into three parts:The first part is the introduction,the second part is the main part of the article,and the third part is the conclusion.The first chapter is the introduction,which introduces the research background and significance of the article.The second chapter is the basic knowledge,introducing some basic definitions and lem-mas.In the third chapter,we study the subrings of graded extensions on K[G,?].The main results are following:Lemma 3.1 Let A=(?)u?GAuXu be a graded extension of V in K[G,?],if u?G and AuXu·A-uX-u=V,then(?)?u?K such that Au=Vau,A-u=V?(-u)(?u-1)=V?-u.Lemma 3.2 Let A=(?)u?GAuXu be a graded extension of V in K[G,?],H={u|AuXu·A-uX-u=V},then H is a group,and for(?)u?H,v?G,there is AuXu·AuXv=Au+vXu+v,AvXv·AuXu=Au+vXu+v.Lemma 3.3 Let A=(?)u?GAuXv be a graded extension of V in K[G,?],H={u|AuXu·A-uX-u=V},HB={u?H|Au=Bu},then HB is a cone of H.Lemma 3.4 For(?)H0 ? H,let#12 then B ? SV(A).Lemma 3.5 LetA=(?)u?GAuXu,B=(?)u?GBuXu,A and B are graded extensions of V in K[G,?],and B(?)A,if Bu(?)Au for a u?G,then BuXu·B-uX-u=V.Let CH be the set of cones of H,which can be used to describe SV(A).We have the following important theorem.Theorem 3.1 Let A=(?)u?GAuXu be a graded extension of V in K[G,?].H={u|AuXu·A-uX-u=V},then there is a one-to-one correspondence between CH and Sv(A).In the fourth chapter,we study the over-rings of graded extensions on K[G,?].The main results are following:Does the over-ring of graded extension A have an upper bound?Lemma 4.1 shows that the union of all elements in QV(A)is still in QV(A).Lemma 4.1 Let A=(?)u?GAuXu be a graded extension of V in K[G,?],#12 then C is a graded extension of V in K[G,?].Let A=(?)u?GAuXu be a graded extension of V in K[G,?],H={u|AuXu·A-uX-u=V},H={u|CuXu·C-uX-u=V},L={u|u?H\H,Au=Cu,A-u?C-u},the set of cones of H containing L and H is denoted as CH+.Theorem 4.1 Let A=(?)u?GAuXu be a graded extension of V in K[G,?],H,L,CH+as described above,then there is a one-to-one correspondence between CH+ and QV(A).In the fifth chapter,we study the cones of Q(n)and the subrings and over-rings of graded extensions on K[Q(n),?].The main results are following:Lemma 5.1 There are only three cones of Q,namely P1={x|x?Q,x?0},P2={x|x?Q,x?0} and Q itself.Lemma 5.2 Let P be a cone of Q(n),for(?)u ? Q(n),u ?P if and only if there is a positive rational number k,which makes ku ? P.Theorem 5.1 Let P be a subset of Q(n)if P is the true cone of Q(n),then there are real numbers a1,a2,...,an that are not all 0,so that(1)for(?)u=(u1,u2,...,un)? Q(n),if a1u1+a2u2+...+anun>0,then u?P.(2)for(?)u=(u1,u2,...,un)? Q(n),if a1u1+a2U2+...+anun<0,thenu(?)P.(3)let S={(u1,u2,...,un)| a1u1+a2u2+...+anun=0},then P ? S is a cone of S.Theorem 5.2 Let a1,a2,...,an be real numbers that are not all 0,and P is a subset of Q(n),S={(u1,u2,...,un)| a1u1+a2u2+...+anun=0,ui?Q,i=1,2,...,n}.if(?)u=(u1,u2,...,un)?Q(n),when a1u1+a2u2+...+anun>0,then u ? P,when a1u1+a2u2+...+anun<0,then u(?)P,and P ?S is a cone of S,then P is a true cone of Q(n).