Torsion classes of extended Dynkin quivers over commutative rings

For a Noetherian $R$-algebra $\mathrm{\Lambda }$, there is a canonical inclusion $\mathsf{t}\mathsf{o}\mathsf{r}\mathsf{s}\mathrm{\Lambda }\to {\prod }_{\mathfrak{p}\in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}R}\mathsf{t}\mathsf{o}\mathsf{r}\mathsf{s}(\kappa (\mathfrak{p})\mathrm{\Lambda })$, and each element in the image satisfies a certain compatibility condition. We call $\mathrm{\Lambda }$ compatible if the image coincides with the set of all compatible elements. For example, for a Dynkin quiver $Q$ and a commutative Noetherian ring $R$ containing a field, the path algebra $RQ$ is compatible. In this paper, we prove that $RQ$ is compatible when $Q$ is an extended Dynkin quiver and $R$ is either a Dedekind domain or a Noetherian semilocal normal ring of dimension two.