Localization respects associated primes for Noetherian rings

Statement
Suppose $$A$$ is a Noetherian commutative unital ring and $$M$$ is any $$A$$-module (not necessarily finitely generated. Let $$S$$ be a multiplicatively closed subset of $$A$$.

There is a natural inclusion on spectra:

$$Spec(S^{-1}A) \to Spec(A)$$

The set of associated primes for $$S^{-1}M$$ as an $$S^{-1}A$$-module is the inverse image in $$Spec(S^{-1}A)$$ of the set of associated primes for $$M$$ as an $$A$$-module.

If we identify $$Spec(S^{-1}A)$$ with its image, a subset of $$Spec(A)$$, then we can write:

$$Ass_{S^{-1}A}S^{-1}M = Ass_AM \cap Spec(S^{-1}A)$$

Proof
The key ingredient in the proof is the fact that if $$m \in M$$, the union of annihilators of all elements of $$Sm$$, can be realized as the annihilator of a single element $$sm$$.