Krull's principal ideal theorem states that in a Noetherian ring $R$, any principal proper ideal $I$ has height at most $1$. Presumably the Noetherian hypothesis is required, so what are some (preferably simple) examples where the result of the theorem doesn't hold?

A search for similar examples turned up this question, but perhaps not requiring $I$ to be prime will provide larger classes of examples, or just some simpler ones.


Let $R=K[X,XY,XY^2,\dots, XY^n,\dots]=K+XK[X,Y]$, and $I=(X)$.

Let $P$ be a prime ideal containing $X$. Then we have $(XY^m)^2=X(XY^{2m})\in P$, thus $(X,XY,XY^2,\dots, XY^n,\dots)\subseteq P$. But $(X,XY,XY^2,\dots, XY^n,\dots)$ is maximal, so $(X,XY,XY^2,\dots, XY^n,\dots)=P$.

We have $(0)\subsetneq P'\subsetneq P$, where $P'=(XY,XY^2,\dots, XY^n,\dots)$. Note that $R/P'\simeq K[X]$, hence $P'$ is prime. Thus the height of $P$ is $\ge 2$ (it's in fact equal to $2$), and therefore the height of $I$ is at least $2$ (it's in fact equal to $2$).


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