Exercise 111.12.1. Let $R$ be a ring, $I \subset R$ an ideal, and $M$ an $R$-module. Compute $\text{depth}_ I(M)$ in the following cases.

$R = \mathbf{Z}$, $I = (30)$, $M = \mathbf{Z}$,

$R = \mathbf{Z}$, $I = (30)$, $M = \mathbf{Z}/(300)$,

$R = \mathbf{Z}$, $I = (30)$, $M = \mathbf{Z}/(7)$,

$R = k[x, y, z]/(x^2 + y^2 + z^2)$, $I = (x, y, z)$, $M = R$,

$R = k[x, y, z, w]/(xz, xw, yz, yw)$, $I = (x, y, z, w)$, $M = R$.

Here $k$ is a field. In the last two cases feel free to localize at the maximal ideal $I$.

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