In a Noetherian ring, the set of associated prime ideals of an ideal is the set of primes which can be written as $(I:z)$.
I'm new to associated primes, and I was wondering why the Noetherian hypothesis here is necessary. In Matsumura's book Commutative Ring Theory, the associated primes of a module $M$ are defined as primes that occur as $\operatorname{ann}(m)$ for some $m\in M$. In this base, a prime $P$ would have to annihilate some element in $A/I$, which means $P$ is the set of elements $p$ such that $px\subset I$ for some $x\in A/I$, which is exactly the definition of $(I:x)$.
What am I missing? Why must we assume $A$ is Noetherian for this to hold?
(The quote comes from @wxu's answer to this question. The question itself has a Noetherian hypothesis. I am trying to figure out why it is necessary.)
Swanson's notes on primary decomposition gives an explanation for this. In particular, go to Remark 3.11 at the bottom of page 9.
For people who don't want to click through the link, here is the gist: Sure, you can define associated primes like that without the Noetherian condition. But there are many other reasonable definitions as well. It happens that for Noetherian rings all these definitions agree so it is more convenient to assume that extra condition.