# 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 $$B_{dR}$$.