Study On Several Properties Of Skew Inverse Laurent Series Rings

The extension of rings plays an important role in the study of Algebra.In recent years,scholars have turned their attention to the ring which is more general,skew inverse Laurent series ring.It mainly contains the following two research directions.Firstly,whether the skew inverse Laurent series ring extensions of some class of rings are still this class of rings;Secondly,it is studied for some properties and structures of skew inverse Laurent series ring.In particular,the second direction has two special cases.That is,(1)? is the identity or ?-derivation ? equals zero,as known as skew-Laurent series ring and pesudo-differential operator ring.(2)the special subring,skew inverse Laurent power series ring.This paper is researched on the first direction and the latter situation of the second direction.We mainly discuss the weak Armendariz property and weak McCoy property of skew inverse Laurent series ring,together with related clean properties of skew inverse Laurent power series ring,such as nil clean property,strongly clean property,uniquely clean property and weakly clean property.This thesis mainly consists of the following components:Chapter 1:We introduce the background,development process and research status of skew inverse Laurent series ring and briefly summarize some important work and results in the literature.Chapter 2:We introduce some essential concepts,such as Abel rings,2-primal rings,(?,?)-compatible rings,(?,?)-SILS Armendariz rings,(?,?)-SILS McCoy rings,clean rings,nil-clean rings and other rings.We also have a list of some results and symbols which are frequently used in the sequel.Chapter 3:We introduce the concept of(?,?)-SILS weak Armendariz rings to study the weak Armendariz property of skew inverse Laurent series rings.Let a be an automorphism of a ring R and ? an ?-derivation of R.Then we prove that if R is weak(?,?)-compatible and nil(R)is a nilpotent ideal,R is(?,?)-SILS weak Armendariz,Tn(R),Sn(R),T(R,n)and T(R,R)are(?,?)-SILS weak Armendariz.Chapter 4:We introduce the concept of(?,?)-SILS weak McCoy rings to study the weak McCoy property of skew inverse Laurent series rings.And we prove that(?,?)-SILS weak Armendariz rings are(?,?)-SILS weak McCoy rings,but the inverse is not true.We also investigate its theoretical properties and the relationship between(?,?)-SILS weak McCoy rings and related rings.Chapter 5:We mainly discuss related clean properties of general skew inverse Laurent power series rings and show some new results:R is a n-clean(feckly reduced,JU,UR,JR,J-boolean)ring if and only if R[[x-1;?,?]]is a n-clean(feckly reduced,JU,UR,JR,J-boolean)ring.Chapter 6:We give a summary about the properties of skew inverse Laurent series rings,and we have made a further outlook about the future research directions.