Question: Compute $\text{Ass}_{\mathbb{Z}} (\mathbb{Z} / n \mathbb{Z})$.
We know that every number has a unique factorization into product of prime numbers, so $n = p_1^{k_1} \cdots p_m^{k_m}$, but it looks not connect to the question. Anyone help me to solve this problem? Thank all!
$\operatorname{Ass}\left(\dfrac{\mathbb{Z}}{n \mathbb{Z}}\right) = \lbrace (p) \in \operatorname{Spec}{\mathbb{Z}}: p\text{ is prime factor of } n\rbrace$.
Indeed, an element $x \in \mathbb{Z}$ is a divisor of zero on $\dfrac{\mathbb{Z}}{n \mathbb{Z}}$ if $\gcd(x, n) \neq 1$, that is, iff $x$ is a multiple of some prime factor $p$ of $n$.