Independently generated modules

A module M over a ring R is said to satisfy (P) if every generating set of M contains an independent generating set. The following results are proved; (1) Let τ = (Tτ, Fτ) be a hereditary torsion theory such that Tτ = Mod-R. Then every τ-torsionfree R-module satisfies (P) if and only if S =R/τ(R) is a division ring. (2) Let K be a hereditary pre-torsion class of modules. Then every module in K satisfies (P) if and only if either K = {0} or S = R/ SocK(R) is a division ring, where SocK(R) = ∩{I ≤ RR: R/I ∈ K}. © 2009 The Korean Mathematical Society.