Torsion classes of extended Dynkin quivers over commutative rings

For a Noetherian ‐algebra , there is a canonical inclusion , and each element in the image satisfies a certain compatibility condition. We call compatible if the image coincides with the set of all compatible elements. For example, for a Dynkin quiver and a commutative Noetherian ring , the path algebra is compatible. In this paper, we prove that is compatible when is an extended Dynkin quiver and is either a Dedekind domain or a Noetherian semilocal normal ring of dimension 2.