Can $A[[X]]$ be a finitely generated $A$-algebra?

Question

Let $X$ be an indeterminate.

Is there a nonzero commutative ring with one $A$ such that $A[[X]]$ is a finitely generated $A$-algebra?

Answer 1

I'll assume AC.

Assume that $A[[X]]$ is finitely generated as an $A$-algebra. I'll derive a contradiction.

Then $A$ has a maximal ideal $M$, and $k=A/M$ is a field. Then $k[[X]]$ is finitely generated over $k$.

This means that $k[[X]]$ is a vector space of countable dimension over $k$. If $k$ is countable, then $k[[X]]$ is uncountable, and so has uncountable dimension over $k$. If $k$ is uncountable, then $1/(1-rX)$ for $r\in k$ form an uncountable linearly independent subset of $k[[X]]$. Again $k[[X]]$ has uncountable dimension over $k$. Thus $k[[X]]$ cannot be finitely generated as a $k$-algebra.