Exercise 17, § 2

Let M be an A-module and N a p-primary submodule in M.

(a) Let F be a submodule of M not contained in N. Show that the transporter N: F is an ideal of A contained in p.

(b) Let a be an ideal of A. Show that if $a \subset N:M$, then $N:a = M$. If $a \not\subset N:M$, N: a is a p-primary submodule in M. If $a \not\subset p$, then $N:a = N$.