The Unique MaximalGF-Regular Submodule of a Module

Abstract

AnR-moduleAis calledGF-regular if, for eacha∈Aandr∈R, there existt∈Rand a positive integernsuch thatrntrna=rna. We proved that each unitaryR-moduleAcontains a unique maximalGF-regular submodule, which we denoted byMGF(A). Furthermore, the radical properties ofAare investigated; we proved that ifAis anR-module andKis a submodule ofA, thenMGF(K)=K∩MGF(A). Moreover, ifAis projective, thenMGF(A)is aG-pure submodule ofAandMGF(A)=M(R)·A.