A characterization of principal rings

Question

I would like to know if the following characterization (for principal rings, not necessarily domains) is true:

$ A $ is principal $ \leftrightarrow $ $ A_\mathscr{M} $ is principal $ \forall \mathscr{M} \in SpecMax (A) $

Where:

$ SpecMax(A) $ is the set of maximal ideals of $ A $.

$ A_\mathscr{M} $ is the localization of $ A $ for the ideal $ \mathscr{M} $

I know it's true that:

$ A $ is principal $ \rightarrow $ $ A_\mathscr{M} $ is principal $ \forall \mathscr{M} \in SpecMax (A) $

I would like to know if anyone knows of any proof for the converse, or some counterexample.

Thank you

Answer 1

Given any commutative von Neumann regular ring, the localizations at maximal ideals are all fields. To make a von Neumann regular ring non-principal, you'd just have to make it non-Noetherian.

So, for example, $\prod_{i=1}^\infty F_2$ would work.