Let $F$ be a number field and $\mathcal O_F$ be its ring of algebraic integers.
Let $(a)$ be a nonzero principal ideal in $\mathcal O_F$ and $I$ be a nonprincipal ideal in $\mathcal O_F$. Note they are just ordinary ideals (ideals that are literally contained in $\mathcal O_F$), not fractional ideals. I wonder if the product $(a)I$ is necessarily non-principal.
More generally one could ask if $IJ$ is a principal ideal in $\mathcal O_F$ then $I$ and $J$ are both principal.
If this is true then I guess this might be true in a more general setting (integral domain ?), so please feel free to point it out if so.