Let A be a Noetherian Ring and M a finitely generated A - module
Recall that for a prime ideal P of A, P is an essential prime of a submodule N of M if and only if for some N$_i$ in the primary decomposition N $= \bigcap N_i$, N$_i$ is P-Primary. The essential primes of the submodule ($0$) are called the associated primes of M.
Prove the following: For any prime ideal p in A, p is an essential prime ideal of some submodule N in M if and only if p contains some p' in the set of associated primes of M
I'm stuck with the proof of this statement.
My first thought was to somehow relate the primary decompositions of N and ($0$) since ($0$) is a submodule of N, but I don't see how to do that (even less so in a way that shows how the relevant prime ideals relate).
I then thought of using the fact that p contains an associated prime of M if and only if it contains ann$_A$M or that M$_p \neq 0$ but I don't see how to bring these back to the primary decomposition of the zero module. I realize my attempts haven't been too fruitful or insightful. I'm very unsure about how to work this out. I'd really appreciate any help with this!