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 ε m and r ε r,there exist t ε r and a positive integer n such that r n t r n x = r n x. 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 m is a gf -regular r module for each maximal ideal of r. many characterizations and properties of gf -regular modules were given. an r -module m is gf -regular iff r / a n n x is a π -regular ring for each 0 ≠ x ε m iff r / a n n m is a π -regular ring for finitely generated module m. if m is a gf -regular module,then j m = 0. © 2013 areej m. abduldaim and sheng chen.