| Popis: |
This work has as its object the study of a rather neglected object, the Lie ring. The general method and type of problem tackled are suggested by analogy with the theory of infinite groups, A recurring theme is the study of residual properties (mainly residual finiteness) of Lie rings, with particular emphasis on soluble ring's. However this by no means presents the whole picture. Related problems in the field of Lie algebras are tackled in the first few chapters, chapters 3, 6, and 7 are not concerned with residual properties at all, and throughout many results are presented for Lie rings which are not necessarily soluble. Ua."1Yof the results (mainly in the second half) will also hold in general nonassociative rings with suitable restrictions imposed, but presentation in this form would make many results which appear natural in the present context seem technical and obscure. Occasional reference is made to general nonassociative rings however. Chapter 1 sets up the notation and a few of the most useful technical tools that are used in the sequel. Chapters 2 and 3 are concerned with certain classes of finitely generated soluble Lie ring (and Lie algebras). The approach is through associative ring theory using the universal enveloping ring. Chapter 2 looks at maximal conditions and residual finiteness while chapter 3 examines the Frattini theory of these Lie rings. Chapter 4 examines the residual properties of certain classes of Lie rings, notably nilpotent Lie rings and Lie rings of matrices over integral domains. Chapter 5 considers Lie rings whose underlying abelian groups satisfy certain rank restrictions. Necessary and sufficient conditions for residual finiteness are established for these rings. In chapter 6 we examine which properties when shared by all the abelian subrings of a soluble Lie ring are inherited by the ring itself. Chapter 7 gives a characterization of certain Lie rings which have the subideal intersection property (i.e. an arbitrary intersection of subideals is once again a subideal). |
