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!



Number of generators of colon and intersection ideals of two finitely generated ideals in a Prufer domain
(For the following we can obviously assume $B$ and $C$ are nonzero.) The equation $BC = (B \cap C)(B+C)$ can be seen by localization. (One of $(B \cap C)\_M$ and $(B+C)\_M$ is $B\_M$ and the other is $C\_M$, depending on which contains which in $D\_M$.) Multiply it by $(BC)^{-1}$ to obtain $D = (B \cap C)(B^{-1}+C^{-1})$. So $(B \cap C)^{-1} = B^{-1} + C^{-1}$ and $B \cap C$ is $(m+n)$-generated. The equation $B \cap C = (B : C)C$ can be seen by localization. (Since $C\_M$ is a principal ideal containing $(B \cap C)\_M$, we have $(B \cap C)\_M = ((B \cap C)\_M : C\_M)C\_M = (B : C)\_MC\_M$. The last equality is using the fact that $C$ is finitely generated.) For each generator of $B \cap C$, write $a = \sum b\_ic\_i$ with the $b\_i$'s in $(B : C)$ and the $c\_i$'s distinct elements from a minimal generating set for $C$. Let $B\_0$ be the ideal generated by the $b\_i$'s, and note that it is $m(m+n)$-generated, since there are $m+n$ sums and at most $m$ $b\_i$'s in each sum. So $B\_0C = B \cap C = (B : C)C$ and $B\_0 = (B : C)$.


I suppose you have probably answered this problem long ago, but I stumbled across this today, and it's related to things I am working on.


