Baer Splitting Beyond $τ_q$-Semisimplicity

For a commutative ring $R$, we call an $R$-module $B$ Baer if $\Ext_R^1(B,T)=0$ for torsion $R$-module $T$. It is known that every Baer module is projective when $R$ is $τ_q$-semisimple, i.e., rings whose total rings of quotients are semisimple. And it was conjectured that the converse characterizes