A Complete Classification Of Graded Extensions In K[Q,σ]

The skew Laurent polynomial rings are an important class of rings.Guangming Xie studied a complete classification of graded extensions in K[X,X^-1;σ],the skew Laurent polynomial ring,and obtained many valuable research results.In[27]and[28],he classified graded extensions in K[X,X^-1,σ],the skew Laurent polynomial ring,by distinguishing eight different types based on the properties of A₁and A_-1,that is,type（a）,type（b）,type（c）,type（d）,type（e）,type（f）,type（g）and type（h）,and described the structures of graded extensions of each type in detail.Let K be a division ring and let V be a total valuation ring of K,that is,for any non-zero k∈K,either k∈V or k^-1∈V.In this paper,we assume that V K.Letσ:Q-→Aut（K）be a group homomorphism and K[Q,σ]be the skew group of Q over K.Assume that K（Q,σ）is left quotient ring of K[Q,σ].Based on the research of graded extensions in K[Z,σ]=K[X,X^-1;σ].In[35],Chunhao Wei classified graded extensions in K[Q,σ]into type（I）and type（II）,and he defined the graded extension of type（a）,type（b）,type（c）,type（d）,type（e）,type（f）,type（g）and type（h）,and dis-cussed the relation of this two classifications and the structures of graded extensions of each type.Unfortunately,although the first classification is a complete classification of graded extensions in K[Q,σ],the second classification is not a complete classification of graded extensions in K[Q,σ].A complete classification of graded extensions in K[Q,σ]will be studied in this article.Like the article of Chunhao Wei,we will classify graded extensions in K[Q,σ]into type（I）and type（II）.Unlike the article of Chunhao Wei,we will redefine the graded extension of type（a）,type（b）,type（c）,type（d）,type（e）,type（f）,type（g）and type（h）,this classification is a complete classification of graded extensions in K[Q,σ].And we will discuss the relation of this two classifications and the structures of graded extensions of each type.Firstly,we will prove that if A=（?）_r∈QA_rX^ris a graded extension of V in K[Q,σ],then A is a graded extension of type（I）if and only if A is a graded extension of type（a）,type（b）,type（c）,type（d）,type（f）or type（g）.Secondly,we will discuss graded extensions of type（II）in K[Q,σ],graded extensions of type（e）and（h）will be studied in detail.Finally,we will give some concrete examples of graded extensions of V in K[Q,σ].This paper is composed of six parts.The first part is the introduction.The second,third,fourth and fifth parts are the main body of this paper.The last part is the conclusion and prospect.The introduction part mainly introduces the research background,research significance and main research results of this paper.In the second chapter,we introduce some basic definitions,lemmas and theorems.In the third chapter,we will give a complete description of graded extensions of Type（I）.In the fourth chapter,we will study graded extensions of Type（II）in detail.In the fifth chapter,we will provide concrete examples of graded extensions of V in K[Q,σ]for illustrating the classification.In the last part,we will give a summary and put forward some conjectures to be tested.