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‐module M is called GF‐regular if for each x ∈ M and r ∈ R, there exist t ∈ R and a positive integer n such that rntrnx = rnx. The notion of G‐pure submodules was introduced to generalize pure submodules and proved that an R‐module M is GF‐regular if and only if every submodule of M is G‐pure iff M𝔐 is a GF‐regular R𝔐‐module for each maximal ideal 𝔐 of R. Many characterizations and properties of GF‐regular modules were given. An R‐module M is GF‐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 module M. If M is a GF‐regular module, then J(M) = 0.