Characterization Of Homocyclic Module Over A Principal Ideal Domain

In this paper,we mainly focus on the characterization of the homocyclic mod-ules over a principal ideal domain.Thus,we give the isomorphic classification of the modules over principal ideal domain.This paper is divided into three parts:The first part is about the introduction,we introduce the research background of this paper and the previous work in this aspect,give the conclusion of this paper.The second part is preparation,we make a list of some basic theory of groups,rings,modules,and structure theorem of the module over principal ideal domain cov-ered in this article.In the third part,we firstly discuss the characterization of three kinds of module over principal ideal domain:finitely generated homocyclic p-module,homogeneous quasicyclic p-module with finite p-rank and divisible torsion-free module with finite rank.If we know the isomorphism of submodule A and B,there exists an automorphis-m of M turn A to B.In turn,we discuss the modules that satisfy this characterization,and the structure of the module is obtained only in these three cases.Thus we can obtain the main theorem in this paper.Theorem 3.10 Let M be a module over principal ideal domain,if to the isomor-phic mapping φ of submodule M1 and M2,there exists an automorphism mapping ψ of M makes φ = ψ|M1.Then if and only if one of the following conditions hold:(1)M is divisible module,each of its primary component of torsion part and torsion-free part has finite rank.(2)M is torsion and each of its primary component is either quasicyclic p-module with finite p-rank or finitely generated homocyclic p-module.At the same time,we do the extension research about the above characterization and give a related theorem.Theorem 3.11 Let M be a finitely generated module over principal ideal domain,M1 and M2 is the submodule of M,and M/M1≌ M/M2.If there exists α∈ AutM makes α induce M/M1 → M/M2.Then if and only if one of the following conditions hold:(1)M is torsion and each of its primary component is finitely generated homo-cyclic p-module.(2)M is free module with finite rank.