Baer Splitting Beyond ${\tau }_q$-Semisimplicity

Abstract

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 ${\tau }_q$-semisimple, i.e., rings whose total rings of quotients are semisimple. And it was conjectured that the converse characterizes ${\tau }_q$-semisimple rings. We show that the converse fails in two substantially different ways. First, the failure is already systematic in the non-reduced Noetherian case. If $D$ is a Dedekind domain which is not a field and $R=D[\epsilon ]/({\epsilon }^2)$, then every Baer $R$-module is projective, although $R$ is not reduced, and thus is not ${\tau }_q$-semisimple. Second, the converse fails even among reduced rings, and in fact among reduced coherent Bézout rings. The main tool is a countable-ring criterion. For a countable Noetherian domain $D$, let $\EC(D)$ be the ring of eventually constant sequences over $D$. If $D$ is not a field, then every Baer $\EC(D)$-module is projective, whereas $\EC(D)$ is reduced and has countably infinitely many minimal prime ideals, and thus is not ${\tau }_q$-semisimple.