INDEPENDENTLY GENERATED MODULES

Abstract

Abstract. A module M over a ring R is said to satisfy ( P ) if every gen-erating set of M contains an independent generating set. The followingresults are proved;(1) Let τ = (T τ , F τ ) be a hereditary torsion theory such that T τ 6=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 everymodule in K satisfies ( P ) if and only if either K = { 0 } or S = R/ Soc K ( R )is a division ring, where Soc K ( R ) = ∩{I ≤ R R : R/I ∈ K} . For a right R -module M , a subset X of M is said to be a generating set of M if M = Σ x∈X xR ; and a minimal generating set of M is any generating set Y of M such that no proper subset of Y can generate M . A generating set X of M is called an independent generating set if Σ x∈X xR = ⊕ x∈X xR . Clearly, everyindependent generating set of M is a minimal generating set, but the converseis not true in general. For example, the set { 2 , 3 } is a minimal generating setof Z