The Torsion-Free Module Of Finite Rank Over A Principal Ideal Domain

The research of Abel group has been a little mature,Abel group can be regarded as Z-module.A natural question is whether these properties and theories can be promoted to the module.With this question the structure and some properties of the torsion-free module of finite rank are discussed in this paper.Torsion module may be regarded as direct sums of cyclic module and quasicyclic module,which means that any torsion module can be represented as the direct sums of some torsion module of rank 1.So it is natural to study the structure of torsion-free module.However,the direct sum and decomposition of torsion module cannot transform to the torsion-free module,which means that the classification of torsion-free module is more complex than the torsion module.This study of this paper summarizes as follows:(1)Firstly,as we know,two torsion-free groups of rank 1 are isomorphism if and only if they have the same type.This article focuses on the special condition of the torsion-free module of rank 1,whose isomorphic invariants,endomorphism ring and automorphism ring are given.Torsion-free module straight and decomposition is discussed,and through an example,we conclude that the torsion-free module of finite rank may have a nonisomorphism direct sum and decomposition.In other words,it is impractical to seek the isomorphism of torsion-free module by the direct sum and decomposition.(2)Next,we discuss three extensions of the torsion-free module of rank 1.It is extended by a module which satisfies maximal and minimal conditions on subgroups and is divisible of finite rank.