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?
2026-03-25 14:07:57.1774447677
A concrete example of an ideal $I\subseteq K[x_1,\ldots,x_n]$ and coprime polynomials $f,g$ such that $(I,f)\cap (I,g)\neq (I,fg)$
64 Views Asked by user228113 https://math.techqa.club/user/user228113/detail AtRelated Questions in POLYNOMIALS
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