## 109.10 Associated primes

Associated primes are discussed in Algebra, Section 10.62

Exercise 109.10.1. 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 109.10.2. 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 109.10.3. 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 $\mathfrak q$.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).