[Math] Find the maximal ideals of the ring $\mathbb{Z}_{36}$.

Question

Find the maximal ideals of the ring $\mathbb{Z}_{36}$.

I don't know where to start on this one.

Any help/hints would be greatly appreciated.

Answer 1

Here is a direct plan of attack to get you started, since I guess the point of the problem is for you to get an understanding of what ideals are rather than use theorems.

What are the subgroups of $(Z_{36}, +)$? Every ideal must be a subgroup of the additive group.
Which of these subgroups are also ideals (that is, they satisfy that if $a\in I$ and $b \in Z_{36}$ then $ab \in I$)?
Order the ideals you get (except the whole ring) by containment. Which of them are not contained in any other (i.e., they are maximal)?