Question

$\textbf{Question: }$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 $n$?

Answer 1

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.