  The aim of this paper is to unify classification theories of torsion classes
of finite dimensional algebras and commutative Noetherian rings. For a
commutative Noetherian ring $R$ and a module-finite $R$-algebra $\Lambda$, we
study the set $\mathsf{tors} \Lambda$ (respectively, $\mathsf{torf}\Lambda$) of
torsion (respectively, torsionfree) classes of the category of finitely
generated $\Lambda$-modules. We construct a bijection from
$\mathsf{torf}\Lambda$ to $\prod_{\mathfrak{p}}
\mathsf{torf}(\kappa(\mathfrak{p}) \otimes_R \Lambda )$, and an embedding
$\Phi_{\rm t}$ from $\mathsf{tors} \Lambda$ to
$\mathbb{T}_R(\Lambda):=\prod_{\mathfrak{p}} \mathsf{tors}(\kappa(\mathfrak{p})
\otimes_R \Lambda)$, where $\mathfrak{p}$ runs all prime ideals of $R$. When
$\Lambda=R$, these give classifications of torsionfree classes, torsion classes
and Serre subcategories of $\mathsf{mod} R$ due to Takahashi, Stanley-Wang and
Gabriel. To give a description of $\mathrm{Im} \Phi_{\rm t}$, we introduce the
notion of compatible elements in $\mathbb{T}_R(\Lambda)$, and prove that all
elements in $\mathrm{Im} \Phi_{\rm t}$ are compatible. We give a sufficient
condition on $(R, \Lambda)$ such that all compatible elements belong to
$\mathrm{Im} \Phi_{\rm t}$ (we call $(R, \Lambda)$ compatible in this case).
For example, if $R$ is semi-local and $\dim R \leq 1$, then $(R, \Lambda)$ is
compatible. We also give a sufficient condition in terms of silting
$\Lambda$-modules. As an application, for a Dynkin quiver $Q$, $(R, RQ)$ is
compatible and we have a poset isomorphism $\mathsf{tors} RQ \simeq
\mathrm{Hom}_{\rm poset}(\mathrm{Spec} R, \mathfrak{C}_Q)$ for the Cambrian
lattice $\mathfrak{C}_Q$ of $Q$.
