The Unique Maximal GF-Regular Submodule of a Module

An R-module A is called GF-regular if, for each a ∈ A and r ∈ R, there exist t ∈ R and a positive integer n such that r (n) tr (n) a = r (n) a. We proved that each unitary R-module A contains a unique maximal GF-regular submodule, which we denoted by M GF(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 M GF(K) = K∩M GF(A). Moreover, if A is projective, then M GF(A) is a G-pure submodule of A and M GF(A) = M(R) · A.