It's clear that $IJ\supset(I+J)(I\cap J)$, but when is the reverse inclusion true?
So far, the simplest counterexample I could find was $I=(x), J=(x^2,y)$ in $k[x,y]$.
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pre-kidney
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for instance, when I and J are co-maximal. – Bobby Dec 30 '13 at 16:47
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Sure, but that's a very specific case. For instance, it looks like in every PID the result holds. – pre-kidney Dec 30 '13 at 16:50