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) $\displaystyle \mathbb{Z}/90\mathbb{Z}$

(b) $\displaystyle \mathbb{Q}[x]/(x^2+1)$

(c) $\displaystyle \mathbb{C}[x]/(x^2+1)$

(d) $\displaystyle \mathbb{Q}[x]/(x^3+x^2)$

My solutions for (a) through (c):

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

(b) Since $\displaystyle x^2+1$ is irreducible in $\displaystyle \mathbb{Q}[x]$, this is a field, so the only maximal ideal is (0).

(c) $\displaystyle \mathbb{C}[x]/(x^2+1) = \mathbb{C}[x]/((x+i)(x-i)) \cong \mathbb{C}[x]/(x+i)\times\mathbb{C}[x]/(x-i) \cong \mathbb{C}\times\mathbb{C}$, 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 $\displaystyle \mathbb{C}\times\mathbb{C}$ to the ideals in the original ring.

For (d), I think it's true that $\displaystyle \mathbb{Q}[x]/(x^3+x^2) \cong \mathbb{Q}[x]/(x^2) \times \mathbb{Q}[x]/(x+1)$, and the second is isomorphic to $\displaystyle \mathbb{Q}$. But what are the maximal ideals of $\displaystyle \mathbb{Q}[x]/(x^2)$?

- Hollywood