Let $A$ be a commutative ring, such that $A^*$ is finite, and that for any maximal ideal $\frak m$, $A/\frak m$ is finite. Is it true that $A$ is denumerable? I've been trying to disprove this by looking at $\mathbf{K}[X]$, where $X=(X_i)_{i \in I}$ and $I$ is non-denumerable but I can't prove that the maximal ideals of this ring are trivial, so that the residue fields would be isomorphic to $\mathbf{K}$ (then choose $\mathbf{K}$ finite). Another idea was looking at maximal ideals of some non-denumerable product $\prod_i K$ but again there may be some strange maximal ideals (like ultrafilters).
What do you think? Maybe the answer to the question is in fact yes?