# Number of generators of colon and intersection ideals of two finitely generated ideals in a Prufer domain

Hi!

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!

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Hi. If it is not much of trouble to you, would you tell me the definition of a Prufer domain, base, and inverse of an ideal? –  Youngsu Apr 14 at 21:50
Sure. When I say "bases," I mean the plural of basis (of an ideal) since I'm considering multiple ideals. A Prufer domain is a domain such that every finitely generated ideal admits an inverse. By an inverse of an ideal $I$, I mean a fractional ideal $I^{-1}$ such that $II^{-1}=R$. –  Reeve Apr 14 at 23:48
Isn't a PID a Prufer domain? If so, how can you have $B \cap C$ minimally generated by more than 1 element? –  Youngsu Apr 15 at 1:21
Yes. PID's are a class of Prufer domains since principal ideals are always invertible. I am not assuming the ring we're working over here is a principal ideal domain, though. Just one where all finitely generated ideals are invertible (not necessarily Noetherian). –  Reeve Apr 15 at 2:23