Structures Of Some Modules Over A Principal Ideal Domain

As an algebraic structure,the issues of the structure and the classification of mod-ules over rings are one of the center questions.For the modules over fields,that is,the vector space over fields,its structures are studied easily,because two finite spaces are isomorphic if and only if they have the same dimensions.However,the structures of the general modules over rings are relatively complex.In this paper,it mainly studies the modules over the principle ideal domain.This is a particular case.The paper consists of three parts.The first part is the introduction;it explains the rationality and significance of the research.The second part is the preparation.As a flagrant contrast,it firstly introduces some decomposition theorems of the abelian group in group theory.Secondly,it gives some basic concepts about rings and modules,such as integral domain,principal ideal domain,module,sub-module,torsion module,quotient module and so on.At last,it introduces some basic theorems of modules over principal ideal domain,mainly on isomorphism of modules.In the third part,it describes some main results.It transplants the methods of decomposition theorem in abelian group(finite or infinite)to the modules over the principal ideal domain,and explores their structures.At last,we get the correspondent decomposition theorems.In this section,specifically,at first,it introduces the structures of the divisible modules and the bounded modules over the principal ideal domain in turn.In other words,it gives the decomposition theorems of the divisible modules and the bounded modules.Among them,it takes the divisible module over Z[i]for example,getting two different structures of modules over Z[i].At last,it proves the structure of the modules with minimal condition and defines the module with minimax condition over the principal ideal domain.Then it portrays the module with minimax condition and illustrates.