Existence of associated prime

Question

For a Noetherian ring $R$, it is well known that $R$ has at least one associated prime. In particular, minimal primes of $R$ are associated primes. My questions is

Question 1: For a commutative ring $R$ with 1 not necessarily Noetherian, is a minimal prime of $R$ an associated prime?

Maybe, more fundamental question is

Question 2: Does a commutative ring with 1 have an associated prime?

I believe the answers are well known. A referece would be appreciated.

user26857 · Accepted Answer · 2016-01-21T20:41:17.350

3

If your definition of associated prime is $\mathfrak p=\operatorname{Ann}(x)$, then the answer is negative to both questions.

The ring $R=K[X_1,\dots,X_n,\dots]/(X_1^2,\dots,X_n^2,\dots)$ doesn't have any associated prime, and, of course, it has a minimal prime ideal.

However, $R$ has a weakly associated prime.

edited Jan 21 '16 at 20:41

answered Jan 21 '16 at 10:08

user26857

52,094

Thank you for the answer. Does it have an "associated prime" under a different definition that is be equivalent if $R$ is Noetherian? – Youngsu Jan 21 '16 at 19:43
@Youngsu See here. – user26857 Jan 21 '16 at 20:33

Existence of associated prime

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