The unique maximal GF-regular submodule of a module

An R-module A is called GF-regular if, for each aA and R, there exist TR and a positive integer n such that rntrna=rna. We proved that each unitary R-module A contains a unique maximal GF-regular submodule, which we denoted by MGF(A). Furthermore, the radical properties of A are investigated; we proved that if A is an R-module and K is a submodule of A, then MGF(K)=K∩MGF(A). Moreover, if A is projective, then MGF(A) is a G-pure submodule of A and MGF(A)=M(R)A.