In an algebra test the following problem was presented:
Any commutative local ring $(R,\mathfrak m)$ with $\mathfrak m$ principal so that $⋂_{i≥0}\mathfrak m^i =0$ is Noetherian and each nonzero ideal of $R$ is a power of $\mathfrak m$.
It could be proven (assuming $R$ to be Noetherian) that each nonzero ideal, being finitely generated, is a subset of a power of $\mathfrak m$. Could anybody be so kind as to give some suggestions. Greateful!