Weak Baer modules over graded rings

In [2], Fuchs and Viljoen introduced and classified the ${\displaystyle B^{\cdot }}$-modules for a valuation ring R: an R-module M is a ${\displaystyle B^{\cdot }}$-module if ${\displaystyle Ex{t}^1\_R(M,X)=0}$ for each divisible module X and each torsion module X with bounded order. The concept of a ${\displaystyle B^{\cdot }}$-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 ${\displaystyle B^{\cdot }}$-modules for a group graded ring. Our most complete result (Theorem 4.10) characterizes ${\displaystyle B^{\cdot }}$-modules for a strongly graded ring R over a finite group G with ${\displaystyle {\left|G\right|}^{-1}\in R}$. Motivated by the results of [8], [9], [10] and [15], we also study the condition that every non-singular R-module is a ${\displaystyle B^{\ast }}$-module with respect to the Goldie torsion theory; for the case in which R is a strongly graded ring over a group, extensive information is obtained for group rings of abelian, solvable and polycyclic-by-finite groups.