I'm wanting to see why the following is true: Given 2 finitely generated ideals $B$ and $C$ in a Prufer domain $D$ with bases of $n$ and $m$ generators respectively, $B\cap C$ has a basis of $m+n$ generators, and $B:C$ has a basis of $m(m+n)$ generators.
This is a result used in Gilmer and Heinzer's paper Overrings of Prufer domains II, Lemma 2, and they use the following facts in a way that is unclear to me: $B:C=B\cap C: C$ (this is clear), so $B\cap C=(B:C)C$, and $BC=(B\cap C)(B+C)$ (I understand the proof for this latter statement), and so since $BC$ is invertible, $B\cap C$ is, and so $(B:C)$ is invertible too.
I understand that if an ideal $A$ has $k$ generators, its inverse has $k$ generators as well, so I think I see how the result on the number of generators of $B:C$ follows from the calculation of the number of generators of $B\cap C$, but I'm not seeing where the number for $B\cap C$ is coming from. How can we make this calculation without knowing the exact number of generators for $BC$ and $B+C$ (we don't know these numbers since we could have redundant generators, I think)?
Does anyone here see what's going on? Thanks!