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
$$|\mathrm{Hom}(M,R)|\cdot|\mathrm{Ext}(M,R)| = 2^{|M|}.$$