Examples of fractional ideals whose inverse does not commute with the product

Question

Let $R$ be an integral domain, $K$ its field of fractions, and $I,J$ fractional ideals.
If $R$ is a Krull domain, then $(R:_KIJ)=(R:_KI)(R:_KJ)$, or $(IJ)^{-1}=I^{-1}J^{-1}$.
But I can't see any reason for this to hold in general, even if $I$ and $J$ are finitely generated. Could anyone provide such an example?

Answer 1

Here's an easy example.

$$ R = k[x,y,z]/\langle x^3+y^3-z^3\rangle $$ $$I = J = \langle x+y,z \rangle.$$

Then $(I^2)^{-1} \cong \langle xz+yz, x^2+2xy+y^2 \rangle$ but $(I^{-1})^2 \cong \langle x^2+2xy+y^2, xz^2 + yz^2, z^4 \rangle$ (instead of writing these as fractional ideals, I'm clearing denominators and embedding them in $R$).

One can easily see that the two ideals are not isomorphic as the second has an associated prime over the origin while the first does not so the original fractional ideals certainly can't be equal.

Of course, up to reflexification / S2-ification, they are the same.