Exercise 13, § 2

(a) Let $u : M \to M'$ be a surjective A-module homomorphism. Show that, if $N'$ is a primary submodule in $M'$, $N = \bar{u}^{-1}(N')$ is primary in M and $r_M(N) = r_{M'}(N')$.

(b) Every intersection of a non-empty finite family of p-primary submodules in an A-module M is p-primary in M. Does the proposition extend to arbitrary intersections?

(c) For an A-module M to be such that {0} is p-primary in M, it is necessary and sufficient that, for every submodule $N \neq 0$ of M, $\operatorname{Supp}(N) = V(p)$. (Note that, for all $x \in M$, M contains a submodule isomorphic to $A/\operatorname{Ann}(x)$.)