I am looking at the maximal ideals of the ring of integers $\mathbb{Z}$ and I see that they are precisely the ideals generated by prime numbers, $(p)$, $p$ prime.
For example, $(2)=\{...,-4,-2,0,2,4,...\}$
$(3)=\{...,-6,-3,0,3,6,...\}$
$(5)=\{...,-10,-5,0,5,10,...\}$
these are indeed maximal: pick one of them, say $(2)$ and suppose $(m)$ for some prime $m$ contains $(2)$, without $(m)$ being $\mathbb{Z}$ itself. Then by looking at the elements of the example ideals above, we see there is no such $(m)$. We cannot have set inclusion of proper ideals of this form, due to their "prime-ness". One ideal does not divide the other, since they are generated by prime numbers, etc.
Now, this leads me to think about the ordering on these maximal ideals. Is there one? I suppose we could apply a relation < given by "m is larger than n" for two ideals $(m)$ and $(n)$. Then $(2) < (3)$, but this doesn't really mean anything.
Can anyone tell me more about order structure on maximal ideals, particularly these ones?