Let R be a ring. We introduce the concept of Gorenstein f-projective R-modules and obtain
some closure properties of Gorenstein f-projective R-modules. As a consequence, we
discuss when every finitely generated R-module has a finitely generated Gorenstein
projective (pre) envelope. For example, we prove that every finitely generated (left and right)
R-module has a finitely generated Gorenstein projective preenvelope if and only if R is a two-
sided coherent ring and M∗= H om R (M, R) has a finitely generated Gorenstein projective …