On Local Modules Hollow Modules and Generalization

Abstract

The main objective of this thesis is to study two concepts in module theory; namely local and hollow modules. There second objective of is to generalize the hollow module. Due to the relations between local and hollow modules, a notion of several properties about both modules. There are some results about local module a pear in section one of chapter two where; every multiplication module is cyclic and hence is local. Also, if the ring R is residually finite, so any module M over is a faithful and locally finite. On the other hand, if M is a cyclic module and is having maximal submodule, then M is a hollow module. Since D1-module is same mearing of lifting module; therefore, every D1-module M over the ring R such that M has an indecomposabilty property; means that to M is also hollow module. Note that every local module is hollow module; then we can say if M is indecomposable projective module over any commutative ring is hollow module. We study the generalization of hollow module in several ways. The first is near maximal, hollow module. If M is a finitely generated and faithful module with N=IM, this imply M is a near-maximal hollow module. Second generalization of hollow module is pure-hollow module. If M is a direct summand of two modules and N≤M, So M is pure-hollow module. Finally, we make the third generalization of hollow module by closed-hollow module. If M is a hollow module and K≤M is relative complement for N≤M, then M is a closed-hollow module.