Ideal Theory Of Gauss Extensions In K(Z^（2）

As an important class of rings,non-commutative valuation rings are of great significance in the research of non-commutative ring theory.Ring extension is an important part of ring theory.At the end of last century,the question of non-commutative valuation ring extensions was put forward by H.H.Brungs,G.Torner and M.Schroder.After that,much important progress has been obtained.Graded extensions and Gauss extensions are two kinds of important valuation ring extensions,and there is a one to one correspondence between graded extensions and Gauss extensions.So it suffices to study ideals of graded extensions in order to study ideals of Gauss extensions.Skew Laurent polynomial ring is an important class of ring.Graded extensions of skew Laurent polynomial ring K[x,x-1;σ]= K[Z,σ]and trivial graded extensions of skew group ring K[Z(2);σ]were discussed in detail by Guangming Xie etc.where Z is the additive group of integers,Aut(K)is the group of automorphisms of a division ring K,and σ is a group homomorphism from Z(2)to Aut(K).But there is no rearch about ideal the of graded extensions of K[Z(2)]and K[Z(2);σ].We will study graded ideals of graded extensions KZ(2)=K[x1,x2’x1-1,and give a complete description of graded prime ideals of graded extensions in KZ(2).In this paper,graded extensions in KZ(2)are given.We will discuss graded ideals of graded extensions in KZ(2)and give a description of graded prime ideals of graded extensions in KZ(2).We will give the corresponding Gauss extensions of these graded extensions,that is,Q(R)Qv(R),SV(R).At last,we will give some examples of each type.This paper is composed of six parts.The first part is introduction.The second,third,fourth,fifth and sixth part are the main body of this paper.And the last part is the concluding remarks.In Chapter 1,some of the research background,research significance,notation,lemmas and propositions,which will be cited in this paper are introduced.In Chapter 2,we will describe graded ideals of graded extensions which is of type(a)in KZ(2)in detail,and give a complete description of graded prime ideals of graded extensions of type(a).The following is the main results:Theorem 2.4 Let A-(?)u∈Z(2)AuXu be a graded extension of type(a)in KZ(2).Then{Ii|i∈△} is the set of all graded prime ideals of A.Ii=(?)u∈Z(2)(?)i αuXu,(?)i∈Ψ1,whereΨ1={(?)i(?)V|i ∈△} is the set of all prime ideals pi of V for an index set △.In Chapter 3,we will describe graded ideals of graded extensions of type(d)in KZ(2)in detail,and give a complete description of graded prime ideals of type(d)graded extensions.The following is the main results:Theorem 3.10 Let A=(?)u∈T WαXu(?)B(?)((?)u∈-T J(W)αuXu)be a graded extension of type(d)in KZ(2),where B=(?)u∈SAuXu,and for any∈S,J(W)αu∈S,J(W)αu(?)Au(?)Wαu.Then {I1iIi∈△}∪{I2j|j=1,2}∪{I3k|k∈Ω} is the set of all graded prime ideals of A.I1i=(?)u∈TWαuXu(?)Ii(?)((?)u∈-T J(W)αuXu),Ii∈Ψ1;I21=(?)u∈TXαuXu(?)((?)u∈Z(2)\T J(W)αuXu);I22=(?)u∈Z(2)J(W)αuXu;I3k=(?)u∈Z(2)(?)kαuXu,(?)k∈Φ3,where Ψ1={Ii(?)B|i∈△} be the set of all graded prime ideals Ψ1 of B with(?)i(?)J(W),for an index set △,(?)i is the set of constants in Ii.T={(x1,x2)∈Z(2)|a1x1+a2x2>0},S={(x1,x2)∈Z(2)|a1+x1+a2x2=0} for not all non-zero a1,a2∈R.Φ3＝{(?)f(?)V|k∈Ω}is the set of all prime ideals(?)k of V with J(W)(?)k,for an index set Ω.In Chapter 4,we will describe graded ideals of graded extensions of type(e)in KZ(2)in detail,and give a complete description of graded prime ideals of graded extensions of type(e).The following is the main result.Theorem 4.11 Let A=(?)u∈Z(2)\S AuXu(?)B be a graded extension of type(e)in KZ(2).Then {I1i|i∈△}∪{I2j|j∈Λ}∪{I3k|kΩ} is the set of all graded prime ideals of A.I1i＝(?)u∈Z(2)\S Wbf(u)αuXu(?)Ii,Ii∈Ψ1;I2j=(?)u∈PjWbf(u)αuXu(?)((?)u∈Z(2))\Pj J(W)bj(u)αauXu),Pj∈∏;I3k=(?)u∈Z(2)(?)kbf(u)αuXu,(?)k∈Φ3,where Φ1={Ii(?)B|i∈△} is the set of all graded prime ideals Ii of B with(?)i(?)J(W),for an index set △,pi isthesetofconstantsinli.∏2＝{Pj(?)Z(2)\S|j∈Λ} is the set of all subsets of Z(2)\S with I =(?)u∈Pj Wbf(u)αuXu(?)((?)u∈Z(2)\Pj J(W)bf(u)αaXu)being graded prime ideals of A,for an index set Λ,Pj may be an empty set.Φ3={(?)k(?)V|k∈Ω} is the set of all prime ideals(?)k of V with J(W)(?)k,for an index set Ω.In Chapter 5,we will describe graded ideals of graded extensions of type(h)in KZ(2)in detail,and give a complete description of graded prime ideals of graded extensions of type(h).The following is main result.Theorem 5.9 Let A=(?)u∈Z(2)\S AuXu(?)B be a graded extension of type(h)in KZ(2)and I=(?)u∈Z(2)IuXu is a idea of A with(?)u∈Z(2),have Au(?)Iu(?)WAu(?),I’=I∩B.Then{I1i|i∈△} is the set of al graded prime ideals of A.Where I1i=(?)u∈Z(2)\S Au(?)iXu(?)Ii,Ii∈Ψ1,where Ψ1＝{Ii(?)B}i∈△} is the set of all graded prime ideals I1 of B,for an index set △,pi is the set of constants in Ii.In Chapter 6,some specific examples of graded prime ideals of graded extensions of each type in KZ(2)will be given.The last part is the concluding remarks.The main work of this paper will be summa-rized.Also some problems and conjectures will be put forward.