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 $n\geq 1$.

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. Is there an example of a local Noetherian ring whose maximal ideal is principle that is not a domain?

2. If so, 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 principle, 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.

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 $n\geq 1$.

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?

2. 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.