Ideals in commutative noetherian rings with unique prime ideal Let $R$ be a commutative noetherian ring with $1$ having only one prime ideal $\mathfrak{P}$. It follows that $\mathfrak{P}^n = 0$ for some integer $n$. Can we say that every proper ideal in $R$ is a power of $\mathfrak{P}$?

Hint:
No. Take a look at $F_2[x,y]/(x,y)^2$ where $F_2$ is the field of two elements.
The answer is affirmative, however, if $\mathfrak{P}$ is principal. Add that hypothesis and try to prove that the other nonzero ideals are powers of it.
