Some Extensions Of Decomposition Theorems In Abelian Groups And Applications

The purposes of this study is to extend as many decomposition theorems in abelian groups as possible to modules over a principal ideal domain(PID)and then using these theorems of modules to study vector spaces and linear transformations on vector spaces and hence obtain decomposition theorems of vector spaces(finite or infinite dimension).This dissertation contains the following three parts:In the first part,several special quasi-cyclic modules over a principal ideal domain are discussed at first;secondly we give the definition of the width of a module and investigate the structure of the modules over a principal ideal domain with finite width according to the different cases of the number of prime elements in the given principal ideal domain,and along this line of thought we discuss respectively the finite dimensional vector space and the infinite dimensional vector space;at last we give a sufficient and necessary condition for a vector space(as a F [?]-module)to be a quasi-cyclic module.In the second part,on the basis of the Pr¨ufer-Baer theorem of the bounded module over a principal ideal domain,we study several basic problems about the algebraic linear transformation of some vector space(infinite dimensional).Let V be a vector space(infinite dimensional)over a field F,A be an algebraic linear transformation of V.1.Suppose any linear transformation commuting with A commutes also with a linear transformation B,then B = f(A),where f is a polynomial over F.2.There exists a basis for V such that the matrix of A relative to this basis has the rational canonical form(classical canonical form).Moreover the classical canonical form becomes the Jordan canonical form when F is algebraic closed.3.There exists the Jordan-Chevalley decomposition of A when F is algebraically closed.This result prevails for the perfect field in general.In the third part,we study the finitely cogenerated module over a principal ideal domain,and give out the decomposition theorem of this kind of modules.Besides,another proof of the decomposition theorem of the finitely generated module over a principal ideal domain is given.