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I am trying to find an example of a noetherian local ring with an associated prime of height greater or equal 1.

That is,

I want a noetherian local ring $R$ together with an associated prime $p$ such that $\text{ht}(p)\geq 1$.

I have tried thinking about examples such as $R = \left(k[x,y]/(y^2,xy)\right)_{(x,y)}$. Here $(y) = ann(x)$ is an associated prime but it has height 0, because $R$ is not a domain. Does anyone know of an example similar to this?

Also, the local hypothesis is somewhat superfluous as one can just find an example in the non-local case, then localise.

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Ok, I didn't think hard enough about my example. In $R = k[x,y]/(y^2,xy)$, $ann(y)$ contains $x$ and $y$ hence $(x,y)\subset ann(y)$ thus $ann(y) = (x,y)$ are the latter is maximal. But then $(y)\subset (x,y)$ and $(x,y)$ and $(y)$ are prime in $R$. Thus $(x,y)$ furnishes an example that I want. (note that we can localise $(k[x,y]/(y^2,xy))_{(x,y)}$ to get a local example).

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