Baer Modules over Domains

Abstract

For a commutative domain $R$ with $1$, an $R$-module $B$ is called a Baer module if $\operatorname {Ext} _R^1(B,T) = 0$ for all torsion $R$-modules $T$. The structure of Baer modules over arbitrary domains is investigated, and the problem is reduced to the case of countably generated Baer modules. This requires a general version of the singular compactness theorem. As an application we show that over $h$-local Prüfer domains, Baer modules are necessarily projective. In addition, we establish an independence result for a weaker version of Baer modules.