Today I learned this interesting fact from Jerry Kaminker: If $A$ is an abelian group such that $\mathrm{Hom}(A,\mathbb{Z})$ and $\mathrm{Ext}(A,\mathbb{Z})$ are both countably generated, then in fact $A$ is finitely generated. This is known in the literature, in some old papers by Nunke-Rotman, Chase, and Mitchell. It makes me interested in possible generalizations.

Suppose that $M$ is a left module over a ring $R$ and that $\mathrm{Ext}^k(M,R)$ is countably generated for all $k$. For which $R$ can you conclude that $M$ is finitely generated or, better, finitely resolved? Any commutative Noetherian ring with finite projective dimension? Is there a countability restriction missing from this proposed generalization? What about non-commutative rings?

The result has been stated for any countable PID rather than just for $\mathbb{Z}$. In fact Mitchell says that if $R$ is a countable PID and $M$ is infinitely generated, then $$|\mathrm{Hom}(M,R)|\cdot|\mathrm{Ext}(M,R)| = 2^{|M|}.$$