Exercise 111.12.6. Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M$ be a finite $R$-module of depth $\geq 2$. Let $N \subset M$ be a nonzero submodule.

Show that $\text{depth}(N) \geq 1$.

Show that $\text{depth}(N) = 1$ if and only if the quotient module $M/N$ has $\text{depth}(M/N) = 0$.

Show there exists a submodule $N' \subset M$ with $N \subset N'$ of finite colength, i.e., $\text{length}_ R(N'/N) < \infty $, such that $N'$ has depth $\geq 2$. Hint: Apply Exercise 111.12.5 to $M/N$ and choose $N'$ to be the inverse image of $K$.

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