Let $d \in \mathbb{Z}$, $d > 1$. Determine all the ideals of $\mathbb{Z}/d\mathbb{Z}$ which are prime or maximal

Question

Let $d \in \mathbb{Z}$, $d > 1$. Determine all the ideals of $\mathbb{Z}/d\mathbb{Z}$ which are prime or maximal

I know that $m\mathbb{Z}/md\mathbb{Z} \cong \mathbb{Z}/d\mathbb{Z}$ as rings without identity if and only if $(d,m)=1$.

I've been stuck on this problem for a good while. Is anyone is able to help me to solve this problem?

Answer 1

Here are some hints:

1. An ideal of a quotient ring $R/I$ has the form $J/I$ where $J$ is an ideal of $R$ containing $I$.

2. The ideals of $\Bbb Z$ are of the form $n\Bbb Z$ for some $n≥0$.

3. The third isomorphism theorem states that $(R/I) \; / \; (J/I) \cong R/J$

4. In a PID (such as $\Bbb Z$), an ideal $I$ is prime iff it is maximal.

Answer:

The prime ideals are exactly of the form $p\Bbb Z / d\Bbb Z$, where $p$ is a prime number that divides $d$. The maximal ideals are also exactly of this form.