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$\textbf{Question: }$We know that the depth of a noetherian local ring is at most the dimension. Do there exist noetherian local rings with high dimension but zero depth? If not, what's the smallest possible depth for a noetherian local ring of dimension $n$?

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up vote 9 down vote accepted

Let $S = k[x_1, \ldots, x_n, y]/(x_1 y, x_2 y, \ldots, x_n y, y^2)$ and let $R$ be the local ring of $S$ at $0$. Then $\dim R = \dim S = n$, but there are no regular elements, since $y$ annihilates the maximal ideal of $R$ - so $R$ has depth zero.

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Great, Thanks! –  LMN Nov 28 '12 at 5:21

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