INDEPENDENTLY GENERATED MODULES

Abstract

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 tau = (T tau,F tau,) be a hereditary torsion theory such that T tau not equal Mod-R. Then every tau-torsionfree R-module satisfies (P) if and only if S = R/tau(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 = Soc(K)(R) is a division ring, where Sock (R) = n{I <= R(R) : R/I is an element of K}.