When does $R [x]/I $ has infinitely many idempotents?

Question

Let $R$ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking for a ring with finitely many idempotents and an unextended ideal $I$ in $R[x]$ such that $R[x]/I$ has infinitely many idempotents.

Thank you for any help.

Answer 1

Let $R=k[y_1,y_2,\dots]$ be a polynomial ring in infinitely many variables over a field and let $I=(x^2y_1^2-xy_1,x^2y_2^2-xy_2,\dots)\subset R[x]$. Then obviously each $xy_n$ is idempotent in $R[x]/I$. Furthermore, the $xy_n$ are all distinct in $R[x]/I$, as can be seen by noting that for any $n$ there is a homomorphism $\alpha_n:R[x]\to k$ such that $\alpha_n(y_n)=\alpha_n(x)=1$ and $\alpha_n(y_m)=0$ for all $m\neq n$, and $I\subset\ker(\varphi_n)$. Finally, $I$ is not extended, and in fact $I\cap R=0$, since clearly $I\subset (x)$.