# Generators for the intersection of two ideals

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

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

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

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

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In a GCD domain you can define a $\mathrm{lcm}$ in which case $I\cap J = \langle\{\mathrm{lcm}(a_i,b_j)\}\rangle$. – JSchlather Sep 13 '12 at 0:59
@JSchlather Your claim is definitely wrong! – user26857 Mar 30 '14 at 19:08
Yes, JSchlather's formula is not correct. $(x,y) \cap (x+y) \neq (x(x+y),y(x+y))$ in $k[x,y]$. – Martin Brandenburg Nov 4 '14 at 21:55

The intersection of any two finitely generated ideals in an integral domain $R$ is also finitely generated if and only if $R$ is coherent. An example of GCD domain which is not coherent can be found in Example 4.4 of this paper. So,