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111.10 Associated primes

Associated primes are discussed in Algebra, Section 10.63

Exercise 111.10.1. Compute the set of associated primes for each of the following modules.

  1. $R = k[x, y]$ and $M = R/(xy(x + y))$,

  2. $R = \mathbf{Z}[x]$ and $M = R/(300x + 75)$, and

  3. $R = k[x, y, z]$ and $M = R/(x^3, x^2y, xz)$.

Here as usual $k$ is a field.

Exercise 111.10.2. Give an example of a Noetherian ring $R$ and a prime ideal $\mathfrak p$ such that $\mathfrak p$ is not the only associated prime of $R/\mathfrak p^2$.

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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