GF-Regular Modules

We introduced and studied GF-regular modules as a generalization of π-regular rings to modules as well as regular modules (in the sense of Fieldhouse). An R-moduleM is called GF-regular if for each x ∈ M and r ∈ R, there exist t ∈ R and a positive integer n such that rntrnx = rx. The notion of G-pure submodules was introduced to generalize pure submodules and proved that an RmoduleM isGF-regular if and only if every submodule ofM isG-pure iffMM is aGF-regular RM-module for each maximal ideal M of R. Many characterizations and properties of GF-regular modules were given. An R-moduleM isGF-regular iff R/ann(x) is a π-regular ring for each 0 ̸ = x ∈ M iff R/ann(M) is a π-regular ring for finitely generated moduleM. IfM is a GF-regular module, then J(M) = 0.