Does the finiteness of $R/P$ imply that of $R/P^r$, where $P$ is a prime ideal?

Question

Conjecture: Let $R$ be a commutative ring with identity. $P$ be a prime ideal of $R$. If $|R/P|<+\infty$, Then $|R/P^r|<+\infty$ for any integer $r\gt 1$.

$|A|$ denotes the cardinality of the set $A$. $R/I$ denotes the quotient ring

I think it does not hold. Could anyone give me a counterexample?

Answer 1

Consider $R=\Bbb F_2[x_n\,:\, n\in\Bbb N]/\langle x_ix_j\,:\, i,j\in\Bbb N\rangle$ and $P=\langle\overline x_n\,:\,n\in\Bbb N\rangle$. $R$ is infinite, $R/P\cong \Bbb F_2$ and $P^2=0$.