Abstract
Let R be a commutative ring with identity and let 𝒬 be the set of finitely generated semiregular ideals of R. A 𝒬-torsion-free R-module M is called a Lucas module if ExtR1(R∕J,M)=0 for any J∈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∝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.
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