Graded Maps Of Q⁽ⁿ⁾ And The Corresponding Graded Extensions

As an important class of rings, non-commutative valuation rings are of great significance in the research of non-commutative ring theory. Recently, the question of non-commutative valuation ring extensions was put forward by H. H. Brungs, G. Torner and M. Schroder, and then many mathematicians have studied it. Because this question is very complicated, so non-commutative ring extensions with some good properties were studied recently. Gauss extensions is a type of non-commutative ring extensions with some good properties. As we knew, there is a one to one correspondence between Gauss extensions and graded extensions. So it suffices to study graded extensions in order to study Gauss extensions. On the other hand, graded maps are very important for us to study graded extensions and the study of themselves is of great value. Let Q be the field of rational numbers, Aut(K) be the set of group automorphisms of a division ring K,σbe a group homomorphism from Q(n) to Aut(K), K[Q(n),σ] be the skew group ring of Q(n) over K, and V be a total valuation ring of K. Assume that K[Q(n),σ] has a left quotient ring K(Q(n),σ). In this paper, we mainly discuss the graded maps over Q(n) and the corresponding graded extensions. Firstly, we give a complete description of graded maps over Q(n). Secondly, we study the corresponding graded extensions in K[Q(n),σ].This paper is composed of five parts. The first part is introduction, the second to fourth part are the main body of this paper, and the last part is the concluding remarks.In introduction, some of the research background, research significance and main results of this paper are introduced.In chapter 1, the complete descriptions of graded maps over Q and graded extensions of type (e) in K[Q,σ] are given. The main results are the following:Theorems 1.5 and 1.10.Theorem 1.5 shows that{fd,fd(1),fd(-1)┃d is a real number} is the set of all graded maps over Q is proved.In Theorem 1.10, the description of graded extensions of type (e) in K[Q,σ] is given:Let W be an overring of V and let A=(?)r∈QArXr be a subset of K[Q,σ] with A0=V. Suppose that J(W)=b-1W for some b∈K. Then A is a graded extensions of Type (e) of V in K[Q,σ]if and only if for all r∈Q,there areαr∈K and a nonzero graded map f with Wαr(αs)σ(r)=Wαr+s((?)r,s∈Q),and Wαr=αrWσ(r),WAr=Wbf(r)αr((?)r∈Q). Furthermore,one of the following properties holds:(1)If either W=V or f(r)+f(-r)=-1((?)r∈Q with r≠0),then A= (?)r∈QWbf(r)αrXr.(2)If W≠V and there is an a∈Q+ with f(a)+f(-a)=0(assume that a is the smallest positive number for this property).Then Ar=Wbf(r)αr((?)r(?)aZ)and B=(?)j∈ZAjaXja is a graded extension of V in K[Xa,X-a,σ(a)]with Wbf(ja)-1αja(?)Aja(?)Wbf(ja)αja((?)j∈Z).The results in this section were published in Journal of Guangxi Normal University (2010,28(2):42-46).In chapter 2,we give a complete description of graded maps over Q(2).Furthermore, we study the corresponding graded extensions over K[Q(2),σ].Let c,d∈R,h be a graded map over Q(2),and let h(i,0)=fc*(i),h(0,j)=fd*(j)for any i,j∈Q.where R is the field of real numbers,fc*∈{fc(1),fc(-1),fc}and fc#∈{fc(1),fc(-1)}.In this chapter,we discuss graded maps over Q(2)in the following cases:(a)c,d∈Q.(ⅰ)h(i,0)=fc(i),h(0,j)=fd(j)for any i,j∈Q.(ⅱ)h(i,0)=fc(i)and h(0,j)=fd#(j)for any i,j∈Q.(ⅲ)h(i,0)=fc#(i)and h(0,j)=fd(j)for any i,j∈Q.(ⅳ)h(i,0)=fc#(i)and h(0,j)=fd#(j)for any i,j∈Q.(b)Either c∈R\Q or d∈R\Q.Because case(ⅱ)is similar to case(ⅲ),so we only discuss case(ⅱ)in the chapter.The main results are the following:Theorems 2.3,2.8,2.9 and 2.12.In Theorem 2.3,the description of graded maps over Q(2)of case(ⅰ)is given.In Theorem 2.8,the description of graded maps over Q(2)of case(ⅱ)or(ⅲ)or(b)is given and another description is given in Theorem 2.9.In Theorem 2.12,the description of the corresponding graded extensions in K[Q(2),σ] is given:Let W be an overring of V and A=(?)u∈Q(2)AuXu be a subset of K[Q(2),σ]with A(0,0)=V. Suppose that J(W)=b-1W for some b∈K and h be a nonzero graded map. Then A is a graded extension corresponding to h of V in K[Q(2),σ]if and only if there areαu∈K\{0)((?)u∈Q(2))with Wαu=αu(W)σ(u),WAu=Wbh(u)αu and Wαu(αυ)σ(u)=Wαu+υ.Furthermore,one of the following properties holds: (1)If either W=V or h(u)+h(-u)=-1 for any u∈Q(2)with u≠(0,0),then Au=Wbh(u)αu.(2)If W≠V and there is an u0∈Q(2)\{(0,0)}with h(u0)+h(-u0)=0,then Au=Wbh(u)αu if h(u)+h(-u)=-1,and B=(?)v∈SAvXv is a graded extension of V in K[S,σ]with Wbh(v)-1αv(?)Av(?)Wbh(v)αv,where s={v▕h(v)+h(-v)=0}.In chapter 3,we give a complete description of graded maps over Q(n).Furthermore, we study the corresponding graded extensions in K[Q(n),σ].Let ci∈R,h be a graded map over Q(n),and let h(0,…,0,r,0,…,0)=fci*(r)for all r∈Q. where the i-coordinate of(0,…0,r,0,…0)is r,other coordinates are 0 (i=1,2,…n).In this chapter,we discuss the graded maps over Q(n)in the following three cases:(a)ci∈Q and fci*=fci for all i(i=1,2,…n).(b)ci∈Q for all i(i=1,2,…n),and there exist i with fci*=fci#.(c)There exist i with ci∈R\Q.The main results are the following:Theorems 3.3,3.11,3.14 and 3.17.In Theorems 3.3,3.11 and 3.14,the descriptions of graded maps over Q(n)of case(a), (b)and(c)are given respectively.In Theorem 3.17,the descripti.on of the corresponding graded extensions in K[Q(n),σ] is given:Let W be an overring of V and A=(?)u∈Q(n)AuXu be a subset of K[Q(n),σ]with A(0,0,…,0)=V. Suppose that J(W)=b-1W for some b∈K and h be a nonzero graded map. Then A is a graded extension corresponding to h of V in K[Q(n),σ]if and only if there areαu∈K\{0}((?)u∈Q(n))with Wαu=αu(W)σ(u),WAu=Wbh(u)αu and Wαu(αv)σ(u)=Wαu+v.Furthermore,one of the following properties holds:(1)If either W=V or h(u)+h(-u)=-1 for any u∈Q(n)with u≠(0,0,…,0),then Au=Wbh(u)αu.(2)If W≠V and there is an u0∈Q(n)\{(0,0,…,0)}with h (u0)+h(-u0)=0,then Au=Wbh(u)αu if h(u)+h(-u)=-1,and B=(?)v∈SAvXv is a graded extension of V in K[S,σ]with Wbh(v)-1αv(?)Av(?)Wbh(v)αv,where S={v▕h(v)+h(-v)=0}.The last part is the concluding remarks.The main work of this paper is summarized, and some problems are put forward.