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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?

  • 3
    In a [GCD domain](http://en.wikipedia.org/wiki/GCD_domain) you can define a $\mathrm{lcm}$ in which case $I\cap J = \langle\{\mathrm{lcm}(a_i,b_j)\}\rangle$.2012-09-13
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    @JSchlather Your claim is definitely wrong!2014-03-30
  • 3
    Yes, JSchlather's formula is not correct. $(x,y) \cap (x+y) \neq (x(x+y),y(x+y))$ in $k[x,y]$.2014-11-04

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