Let $A$ be an integral domain and $I$, $J$ be non-zero ideals. Is $((I \cap J)^{-1})^{-1} = (I^{-1})^{-1} \cap (J^{-1})^{-1}$?
For an ideal $I$, we define $I^{-1} = \{x \in K \mid xI \subset A\}$, where $K$ is the field of fractions of $A$.
The motivation came from van der Waerden's Algebra, the section 105: the ideal theory on a Noetherian integrally closed domain. See here.
For an integrally closed domain $D,\,$ the $v$-operation $\, I\mapsto I^v = (I^{-1})^{-1}$ distributes over finite intersections of finitely generated fractional ideals iff $D$ is a $v$-domain, see Prop. $6.1$ in M. Fontana and M. Zafrullah, On v-domains: a survey.
$v$-domains are generalizations of Krull and Prufer domains. In your special case of Noetherian domains we have that a domain is a $v$-domain $\iff$ integrally-closed $\iff$ Krull.
