Recall that an epimorphism $f : A \to B$ in a category with fiber products is called stable (or universal) if for every morphism $C \to B$ the base change $A \times_B C \to C$ is an epimorphism.
Question. Is there a classification of stable epimorphisms in the category of commutative rings? If not, what can we say about them? What about examples?
Clearly all surjective homomorphisms are stable. I have looked at the "classical" example of a non-surjective epimorphism, the inclusion $\mathbb{Z} \to \mathbb{Q}$, but couldn't find out if it is stable. If $C \to \mathbb{Q}$ is injective, then $C$ is a localization of $\mathbb{Z}$ and it follows that $\mathbb{Z} \cong \mathbb{Z} \times_{\mathbb{Q}} C \to C$ is an epimorphism. If $C$ is a $\mathbb{Q}$-algebra, then $\mathbb{Z} \times_{\mathbb{Q}} C \to C$ is the localization at $\mathbb{Z} \setminus \{0\}$ and hence, again, an epimorphism. But I don't know what happens for arbitrary homomorphisms $C \to \mathbb{Q}$.
PS: A reference for the general theory of epimorphisms of commutative rings is: Séminaire Samuel. Algèbre commutative, 2, 1967-1968: Les épimorphismes d'anneaux, available on nundam. See also this mathoverflow discussion.