Let be a quasi-local domain (i.e. a commutative domain with finitely many maximal ideals). Suppose that for every maximal ideal of , the localization is a P.I.D. Show that is a P.I.D.
Let be a quasi-local domain (i.e. a commutative domain with finitely many maximal ideals). Suppose that for every maximal ideal of , the localization is a P.I.D. Show that is a P.I.D.
I'm still thinking about this but here's my approach. First consider only local domains for which there's only one maximal ideal. Let be an ideal. Then the expansion of to is principal, by assumption. Let be its generator, where and . Now is a candidate for the generator of but now to prove it. Obviously the maximality of is critical.
OK, I think I have a proof. It may not be the most direct, however. We know that any local domain who's maximal ideal is principal must be a discrete valuation ring. Now, let $X$ be the maximal spectrum of $A$. Then,
Since each is a DVR and the intersection is finite, we have that is a PID.