TITLE: Computing intersection of subrings
QUESTION [13 upvotes]: Let $R$ be a finitely generated commutative ring over a field, for concreteness.

If $S,T \leq R$ are two finitely generated subrings, is their intersection
  also finitely generated?

(Certainly this isn't true for infinite intersections.)
If so,

How can one compute such an intersection using a computer algebra package?

It's easy to reduce to the case that $R = {\mathrm k}[s_1,\ldots,s_n,t_1,\ldots,t_m]/I$, the subring $S$ is generated by $s_1,\ldots,s_n$, and $T$ is generated by $t_1,\ldots,t_m$.

REPLY [4 votes]: A couple of years ago, David Speyer showed me the following counterexample.
Let $E$ be an elliptic curve defined over a field, with points $P$, $Q$ that are linearly independent under the group law. Let $p:X\to E$ be the total space of the rank two vector bundle $\mathcal{O}([P])\oplus\mathcal{O}(-[Q])$ on $E$. Then $X$ is finite type, but it can be checked that the ring $\Gamma(\mathcal{O}_X)$ of global regular functions on $X$ is not finitely generated.
If we pick an affine open cover $\{U,V\}$ of $E$, then $\{p^{-1}(U),p^{-1}(V)\}$ is an affine open cover of $X$. The rings of regular functions on $p^{-1}(U)$ and $p^{-1}(V)$ are finitely generated, but their intersection inside the function field of $X$ (or inside the finitely generated subalgebra they generate) is $\Gamma(\mathcal{O}_X)$, which is not finitely generated.