I seem to have difficulty with this type of problem - perhaps a lack of intuition about prime and maximal ideals. Here's the problem. I'm having trouble with part (d).

For each of the following rings, list all maximal ideals.

(a)

(b)

(c)

(d)

My solutions for (a) through (c):

(a) These are generated by the prime factors of 90: (2), (3), and (5).

(b) Since is irreducible in , this is a field, so the only maximal ideal is (0).

(c) , so the maximal ideals are ((0,1)) and ((1,0)), which correspond to the ideals (x+i) and (x-i) in the original ring.

I think that answer is correct, but I'm not entirely clear on how I got from the ideals in to the ideals in the original ring.

For (d), I think it's true that , and the second is isomorphic to . But what are the maximal ideals of ?

- Hollywood