Ideal Theory Of Gauss Extensions In K(Z;?)

Skew group ring is an important class of rings,and recently,it has been researched by many mathematicians at home and abroad.H.Marubayashi and Gangming Xie had made a classification of graded extensions in K[Z,?]including type(a),type(b),type(c),type(d),type(e),type(f),type(g)and type(h).H.H.Brungs,H.Marubayashi and E.Osmanagic had researched the primes of Gauss extensions over crossed product algebras.So it is valuable to study the ideal theory of graded extensions in K[Z;?].Based on the researches of H.H.Brungs,H.Marubayashi,E.Osmanagic,Guangming Xie,etc.,ideal theory of graded extensions in K[Z;?]will be discussed in this paper.This thesis is divided into five sections:introduction,the main body being composed of three chapters and the conclusion.The main results of this thesis will be given in the three chapters.In the introduction,the research background,research significance and the main results of this thesis are introduced.In Chapter 1 of this thesis,some basic properties and basic concepts in skew group rings will be introduced.In Chapter 2 of this thesis,we mainly discuss ideal theory of graded extensions of type(a),type(b),type(c),type(d)and type(e).Particularly,the prime ideas and the completely prime ideas of the graded extensions of type(b)in K[Z;?]will be researched in detail,and the results can be concluded as follow:Let W be an overring ring of V,A=(?)i<0?i?i(J(W))Xi(?)V(?)((?)i>0W?iXi)ba a graded extension of V in K[Z;?(type(b)).Then{I1m|m??}?{I2n|n??}?{I}?{J}is the set of all graded prime ideas of A,where I1m=(?)i<0?i?i(J(W))Xi(?)m(?)((?)i>0W?1Xi),(?)m?Pm.I2n=(?)i?Z(?)n?iXi,(?)n?Pn.I=(?)i?0?i?i(J(W))Xi(?)((?)i>0W?iXi).J=(?)i?0?i?i(J(W))Xi(?)((?)i>0J(W)?iXi).A prime idea I1m,I2n of A is completely prime idea if and only if the corresponding prime idea(?)m,(?)n is completely prime idea of V.In Chapter 3 of this thesis,we mainly discuss ideal theory of graded extensions of type(f),type(g)and type(h).Particularly,the prime ideas and the completely prime ideas of the graded extensions of type(g)in K[Z,?]will be researched in detail,and the results can be concluded as follow:Let W be an overring ring of V,A =(?)i?ZMiXi ba a graded extension of V in K[Z,?](type(g)).Then {I=(?)i?Z?Xi|Mi?i(?)(?)?(?)Mi,T is the prime idea of V} is the set of all graded prime ideas of A.The last part is concluding remarks.The main work in this paper will be summarized and some unsolved questions will be proposed.