How to find the ideals of $\Bbb{Z}_n$

Question

I have a homework problem to find the maximal ideals in $\Bbb{Z}_8$, $\Bbb{Z}_{10}$, $\Bbb{Z}_{12}$, and $\Bbb{Z}_n$. That question has already been asked on here, but I don't even understand how to find just plain ideals of $\Bbb{Z}$.

Answer 1

Consider the canonical map $\pi : \mathbb Z \to \mathbb Z_n$. This is a surjective ring homomorphism and so the ideals of $\mathbb Z_n$ correspond to the ideals of $\mathbb Z$ that contain $\ker \pi= n\mathbb Z$.

All ideals of $\mathbb Z$ are of the form $m\mathbb Z$. (Because ideal and subgroup is the same for $\mathbb Z$.)

Finally, $m\mathbb Z$ contains $n\mathbb Z$ iff ...