Subrings And Overrings Of Graded Extensions In Kz^{(n)}

Skew group rings are one of the most important rings in modern algebra.Graded extensions and Gauss extensions are two types of important ring extensions.Therefore,it is very important to study the graded extensions on the skew group rings and corresponding Gauss extensions.Skew Laurent polynomial rings are very important rings.From special to general,through the research of graded extensions of skew Laurent polynomial rings further to graded extensions of skew group rings.In recent years,many results are attained on graded extensions in skew Laurent polynomial rings by Professor Guangming Xie and H.Marubayshi etc.They divided the graded extensions in skew Laurent polynomial rings according to the properties of A1 and A-1,into eight different types,that is,type(a),type(b),type(c),type(d),type(e),type(f),type(g),type(h).similarly,graded extensions in skew group rings KZ(n)can be divided into four different types.In this paper,based on the research of graded extensions of KZ(n),we mainly study subrings and overrings of graded extensions in KZ(n).Based on the structure of graded extensions of KZ(n),we use the theory of cone to study subrings and overrings of graded extensions in KZ(n).We will show that there is a one to one correspondence between the set of subrings(overrings)of a graded extensions in KZ(n)and the set of cones of corresponding group.Finally,subrings and overrings of graded extensions in KZ(n)are discussed,and some corresponding examples are given.This paper is composed of three parts.The first part is the introduction.The second part is the main body of this paper.The third part is the conclusion.The research background and significance of this paper are introduced in the preface.In the first chapter,we introduce some basic definitions,lemmas,and the structures of the graded extensions in KZ(n).In the second chapter,we begin to discuss subrings of graded extensions on KZ(n).The main results are the following:Theorem 2.1 Let A =(?)u∈Z(n)AuXu be a graded extension of V in KZ(n),H = {u∈ Z(n)AuXuA-uX-u = V}.Then there is a one-to-one correspondence between Sv(A)and CH(CH is the set of cons of H).Let R(A)be the rank of H above.Proposition 2.1 Let A=(?)u∈Z(n)AuXu be a graded extension of V in KZ(n).If R(A)=0,then Sv(A)={A}.Proposition 2.2 Let A=(?)u∈Z(n)AuXu be a graded extension of V in KZ(n).If R(A)=1,then|Sv(A)|=3.Proposition 2.3 Let A=(?)u∈Z(n)AuXu be a graded extension of V in KZ(n).If R(A)=m≥2.then |Sv(A)|=3,here |Sv(A)| is the cardinal number of Sv(A).In the third chapter,we study overrings of graded extensions in KZ(n).The main results are the following:Let A=(?)u∈Z(n)AuXu be a graded extension of V in KZ(n),Q-1=|q<0|,Z-={z ∈ Z|z<0}.Let C = ∪B∈Qv(A)B=u∈Z(n)CuXu,H= {u|Au·A-u=V},M= {u∈Z(n)|Au =Vαu,A-u= J(V)αu-1,there exist B=(?)u∈Z(n)\Zu AvXvv(?)((?)l≥0AluXlu)(?)((?)m>0 V(αu-1)mX-mu)=(?)u∈Z(n)BuXu∈Qv(A)}.Let H be the subgroup of Z(n)which is spanned by H and M,CH={H0|H0 is a cone of H,H0(?)H,H0(?)M}.Theorem 3.2 Let A =(?)u∈Z(n)AuXu be a graded extension of V in KZ(n).CH is defined as above,then there is a one-to-one correspondence between CH and Qv(A).Proposition 3.1 If R(H)=R(H)+1,then |Qv(A)|=2.Proposition 3.2 If A(?)B,then R(HA)<R(HB).Proposition 3.3 If R(H)=R(H)+r,then |Qv(A)|≤r+1.In the fourth chapter,subrings and overrings of graded extensions in KZ(n)are discussed,and some corresponding examples are given.In the last part,we mainly summarize the research works of this paper,and put forward some extended research problems.