commutative algebra – Quotient by powers of a principal maximal ideal is Artinian

Let $R$ be a commutative ring (if necessary we assume it is an integral domain), and $mathfrak{a}=(f)$ be a maximal ideal that is principal. Is it true that $R/mathfrak{a}^n$ is a local Artinian for all $n>0$?

I can see that it is local since there can not be any other maximal ideal lying over $mathfrak{a}^n=(f^n)$. To show it is Artinian, is it possible to show $mathfrak{a}^m/mathfrak{a}^n$ are the only possible ideals?

I was reading this note on page 60 about the $p$-adic period ring $B_{dR}$.