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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