Bulletin of the
Korean Mathematical Society BKMS

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 $\tau=(\mathbb{T}_{\tau},\mathbb{F}_{\tau})$ be a hereditary torsion theory such that $\mathbb{T}_{\tau} \neq$ {\rm Mod}-$R$. Then every $\tau $-torsionfree $R$-module satisfies $(P)$ if and only if $S=R/\tau (R)$ is a division ring. $(2)$ Let $\mathcal K$ be a hereditary pre-torsion class of modules. Then every module in $\mathcal K$ satisfies $(P)$ if and only if either ${\mathcal K}=\{0\}$ or $S=R/{\rm Soc}_{\mathcal K}(R)$ is a division ring, where ${\rm Soc}_{\mathcal K}(R)=\cap \{I \leq R_R: R/I\in {\mathcal K}\}$.

Keywords: generated set for modules, basis, (non)-singular modules, division ring, torsion theory

MSC numbers: 16D10