Let $R$ be a Noetherian local domain with unique maximal ideal $M$. Then I want to show that if every $M$-primary ideal is a power of $M$, then $R$ is a Discrete Valuation Ring.
I know I'll be done if I can show that $M$ is principal, or that $M$ is the only prime ideal (since then I can invoke, or that $R$ is integrally closed in its field of fractions, but I'm not sure how to show any of those things. Could I have some hints?
There is no ideal (properly) between $M^2$ and $M$ (why?). Let $x\in M-M^2$. (What can you say if $M=M^2$?) Then $M^2+(x)=M$, and from Nakayama Lemma get $M=(x)$.