The statement "a local ring whose maximal ideal is principal is Noetherian" is (I think) false. The ring of germs about $0$ of $C^\infty$ functions on the real line seems to be a counterexample since $e^{-1/x^2}\in \left(x^n\right)$ for all $n\geq 1$.

If I add to the hypothesis that the ring is a domain, then (I think) the statement is true. I'm trying to figure out if this

*must*be true (I suspect not). Is there a nice example of a local Noetherian ring whose maximal ideal is principal that is not a domain?Is there a better, weaker condition to add to the hypothesis so that sufficiency holds? In other words, "

*if R is a local ring whose maximal ideal is principal, then R is Noetherian if and only if R is*[what is the best thing to put here]*?*"

Here local rings are assumed to be commutative with unity.