The Stacks project

Exercise 111.10.3. Let $R$ be a Noetherian ring with incomparable prime ideals $\mathfrak p$, $\mathfrak q$, i.e., $\mathfrak p \not\subset \mathfrak q$ and $\mathfrak q \not\subset \mathfrak p$.

  1. Show that for $N = R/(\mathfrak p \cap \mathfrak q)$ we have $\text{Ass}(N) = \{ \mathfrak p, \mathfrak q\} $.

  2. Show by an example that the module $M = R/\mathfrak p \mathfrak q$ can have an associated prime not equal to $\mathfrak p$ or $\mathfrak q$.

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View Exercise 111.10.3 as pdf