Automorphisms Of Modules Over A Principal Ideal Domain

The lifting problem of isomorphism is profound,complex and active in the basic theory of algebra.This paper is about the lifting problem of isomorphism and it is based on the module over principal ideal domain.We can find an automorphism of this module to let two elements of this module map one to the other.We also investigate the dual form of the above automorphism,then we get a similar conclusion.In this paper,we find an efficient method of calculating elementary divisors of arbitrary finitely generated tor-sion module,the structure of which is uniquely determined by elementary divisors.This paper is divided into the following four parts:The first section is about the introduce of simple history of abstract algebra and the background of this paper,also about the significance of the automorphism of the module over principle ideal domain.The second part is preparation.It introduces the principal ideal domain,automor-phism,module,torsion module,finitely generated module,free module,cyclic module,locally cyclic module,elementary divisor,annihilator and some basic theory.The third part is about the extend of the automorphism of modules over principal ideal domain which satisfy the maximal condition.It shows the conclusion of the ex-tend of the modules over principal ideal domain which satisfy the maximal condition by discussing the the extend of the automorphism of finitely generated torsion modules and finitely generated modules over principal ideal domain.The fourth part is on the base of the third part.It shows a dual conclusion of the extend of the automorphism of modules over principal ideal domain which satisfy the maximal condition,which is the conclusion of the extend of the modules over principal ideal domain which satisfy the minimal condition.