I know that in a polyomial ring $K[x_1,\ldots,x_n]$ over a field $K$, given a monomial ideal $I$ and two coprime monomials $f,g\notin I$, it holds $$(I,f)\cap (I,g)=(I,fg)$$ However, I've been unproductively trying to find an explicit case where the identity fails and one of the three ideals $I, (f), (g)$ is not monomial ($f,g$ still coprime, of course). Can someone provide a classic example, or a link, or a way to find it other than fishing for a lucky guess?
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Do you have a source for this theorem? – alphacapture Jul 03 '16 at 23:48
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@alphacapture you use modular law (which holds for triplets of monomial ideals) twice. – Jul 03 '16 at 23:51
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3What if I take $I = (y)$, $f = x - y$, $g = x + y$? I think it's a nice picture. – Hoot Jul 03 '16 at 23:54
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Well, there are those times when you assume everything is complicated! Thank you all. – Jul 04 '16 at 00:00