In [2], Fuchs and Viljoen introduced and classified the $B^{*}$-modules for a valuation ring R: an R-module M is a $B^{*}$-module if $Ex{t}_R^1(M,X)=0$ for each divisible module X and each torsion module X with bounded order. The concept of a $B^{*}$-module was extended to the setting of a torsion theory over an associative ring in [14]. In the present paper, we use categorical methods to investigate the $B^{*}$-modules for a group graded ring. Our most complete result (Theorem 4.10) characterizes $B^{*}$-modules for a strongly graded ring R...

Recently, A. Facchini [3] showed that the classical Krull-Schmidt theorem fails for serial modules of finite Goldie dimension and he proved a weak version of this theorem within this class. In this remark we shall build this theory axiomatically and then we apply the results obtained to a class of some modules that are torsionfree with respect to a given hereditary torsion theory. As a special case we obtain that the weak Krull-Schmidt theorem holds for the class of modules that are both uniform...

Let Fl(R) denote the category of flat right modules over an associative ring R. We find necessary and sufficient conditions for Fl(R) to be a Grothendieck category, in terms of properties of the ring R.