Module-theoretic characterizations of the ring of finite fractions of a commutative ring

Abstract

Let $R$ be a commutative ring with identity and let $\mathcal{𝒬}$ be the set of finitely generated semiregular ideals of $R$. A $\mathcal{𝒬}$-torsion-free $R$-module $M$ is called a Lucas module if ${\text{Ext}}_R^1(R∕J,M)=0$ for any $J\in \mathcal{𝒬}$. Moreover, $R$ is called a DQ ring if every ideal of $R$ is a Lucas module. We prove that if the small finitistic dimension of $R$ is zero, then $R$ is a DQ ring. In terms of a trivial extension, we construct a total ring of quotients of the type $R=D\propto H$ which is not a DQ ring. Thus in this case, the small finitistic dimension of $R$ is not zero. This provides a negative answer to an open problem posed by Cahen et al.