Minimal prime ideals of the support of a module

Question

in a book that I am reading it is claimed (without proof) that all minimal prime ideals of the support of a finite module $M$ over a local Noetherian Ring forms a subset of the set of all associated primes, Ass $M$. Is this true in general of we remove the assumption that the ring is local? I have a hunch that it is. Thank you.

Answer 1

It is indeed true, and $M$ needs not be finitely generated. It is a direct corollary of this result:

Let $A$ be a (commutative) ring, $M$ an $A$-module.

(i) Any prime ideal containing an element of $\operatorname{Ass}M$ belongs to $\operatorname{Supp}M$.

(ii) Conversely, if $A$ is noetherian, any prime ideal in $\operatorname{Supp}M$ contains an element of $\operatorname{Ass}M$.

Reference: N. Bourbaki, Commutative Algebra, Ch. IV, Associated Prime ideals and Primary Decomposition*, § 1.3, proposition 7.