Independently Generated Modules

Abstract

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 τ = (6τ , .τ ) be a hereditary torsion theory such that 6τ = 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 eitherFor 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, every independent generating set of M is a minimal generating set, but the converse is not true in general. For example, the set {2, 3} is a minimal generating set of Z Z but not an independent generating set.It is well-known that every generating set of a right vector space over a division ring contains a minimal generating set (or a basis). This motivated various interests in characterizing the rings R such that every module in a certain class of right R-modules contains a minimal generating set, or every generating set of each module in a certain class of right R-modules contains a minimal generating set (see, for example, In [2, Theorem 2.3], the authors proved that R is a division ring if and only if every R-module has a basis if and only if every irredundant subset of an R-module is independent. This result can be considered in a more general context of a torsion theory. For an R-module M , M is said to satisfy (P ) if every