Proposition 4.2 of P. Samuel's "Lectures on UFD's" says that the localization of a Krull ring $A$ with respect to a multiplicatively closed set $S$, $S^{-1}A$, is again a Krull ring.
My question: When the converse of the above proposition is true?
Generally there exists a counterexample; just take any integral domain $A$ which is not a Krull ring and $S=A-\{0\}$. Then $S^{-1}A$ is the field of fractions of $A$, which is trivially a Krull ring, but $A$ is not a Krull ring (by assumption).
Therefore: What additional conditions on the multiplicatively closed set $S$ and/or on $A$ one must impose in order to get the converse? For example, is it true that assuming that $S$ is "primal" will imply that $A$ is Krull (given $S^{-1}A$ is Krull, of course)? See Page 251; Theorem 15.39 (Nagata's theorem). What about $S$ generated by one element? (not necessarily a prime element).