generators of intersection of ideals

Question

Let $I=\langle a_1,\cdots, a_s\rangle, J=\langle b_1,\cdots, b_t\rangle$ be ideals of arbitrary commutative ring.

Then we know that $I+J=\langle a_1,\cdots, a_s, b_1,\cdots, b_t\rangle, IJ=\langle\{a_ib_j | 1 \leq i \leq s, 1\leq j \leq t\}\rangle$.

Also $IJ\subseteq I\cap J \subseteq I+J$.

Studying, I wonder the generators of $I\cap J$. Is it possible that know generators? Or is it finitely generated?