Bull. Korean Math. Soc. 46 (2009), No. 5, pp. 867–871 DOI 10.4134/BKMS.2009.46.5.867 INDEPENDENTLY GENERATED MODULES Muhammet Tamer Kos ¸an and Tufan ¨ Ozdin 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 following results are proved; (1) Let τ =( τ , τ ) be a hereditary torsion theory such that τ = 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/ 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, 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, [2], [8], [9], [11]). 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 Received July 13, 2008. 2000 Mathematics Subject Classification. 16D10. Key words and phrases. generated set for modules, basis, (non)-singular modules, division ring, torsion theory. The content of this paper is a part of a thesis written by Tufan ¨ Ozdin under the supervision of M. Tamer Ko¸ san (Gebze Institute of Technology). c 2009 The Korean Mathematical Society 867