Prime and Maximal Ideals of $\mathbb{Z}_2 \times \mathbb{Z}_4$ I'm having some trouble finding ideals in general. The problem I'm stuck on is:

Find all prime and maximal ideals of $\mathbb{Z}_2 \times \mathbb{Z}_4$. 

I know that a finite integral domain is a field, which means the prime and maximal ideals are the same. That means I just have to find one or the other, right? But I don't really know how to go about finding ideals for $\mathbb{Z}_n \times \mathbb{Z}_m$. Any help would be great. Thank you!
An ideal of $Z_2 \times Z_4$ must have the form $I \times J$ where $I$ is an ideal of $Z_2$ and $J$ is an ideal of $Z_4$, both of which are not necessarily proper. So clearly there's not so much ideals to try!
An ideal of $Z_2 \times Z_4$ must have the form $I \times J$ where $I$ is an ideal of $Z_2$ and $J$ is an ideal of $Z_4$, both of which are not necessarily proper. So clearly there's not so much ideals to try!