Kothe Radical Of Multiplication Modules

Let R be a commutative ring with identity and M a unital R-module. The module M is said to be a multiplication module provided for each submodule N of M there exists an ideal I of R such that N=IM. Multiplication modules are a "young" class of modules. It was introduced by A. Barnad in1981. In1988, Z. A. El-Bast and P.F. Smith proved an R-module M is a multiplication module if and only if M is P-torsion or P-cyclic for every maximal ideal P of R and investigated some properties of multiplication modules without using the method of localization. After that, multiplication modules have been studied extensively. Hitherto there are mainly three research directions about multipli-cation modules that the authors are aware of:(i) Submodules of multiplication modules;(ii) Generalizations and dualizations of multiplication modules;(iii) Endomorphisms of multiplication modules.This paper is mostly motivated by the recent works in the first and third directions. It mainly investigates the nilpotence in multiplication modules. Then we define and characterize the Kothe radical of a multiplication module and finally investigate the re-lationship between multiplication modules and (co-)Hopfian modules.This paper consists of three sections.In the first section, we briefly review some definitions, notations and auxiliary results.In the second section, we firstly extend the nilpotent and nil ideals of rings to multi-plication modules by defining the nilpotent and nil submodules of multiplication modules and then investigate some properties of the nilpotent and nil submodules of a multipli-cation module. Then we define and characterize the Kothe radical of a multiplication module. The equivalence between Kothe radical, the sum of nil submodules and the intersection of prime submodules is proved. Finally we obtain some related results of Kothe semisimple and radical modules.In the last section, we firstly investigate the construction of the endomorphism rings of finitely generated modules over non-commutative rings and obtain the fact that the endomorphism ring of a finitely generated left R-module is the homomorphic image of a subring of the matrix ring over R. Then we investigate the construction of the endo- morphism rings of finitely generated multiplication modules. Finally we investigate the relationship between multiplication modules and (co-)Hopfian modules.