User:J D Bowen/Math899 HW3

2.6.3) $\mathbb{Z} \ $ is a principal ideal domain, so the ideals are simply $\left\{{\langle n \rangle : n\in\mathbb{Z} }\right\} \ $ and the prime ideals are maximal.

Suppose we have for any three ideals $\langle a \rangle, \langle b \rangle, \langle c \rangle \ $,

$\langle a \rangle \langle b \rangle \subset \langle c \rangle \ $.

We know $c|a\land c|b \iff c|ab $ if c is prime.

If c is not prime, this will not be true; hence, the prime ideals are precisely $\left\{{\langle p \rangle : p \ \text{prime} }\right\} \ $. So this is $\text{maxSpec}(\mathbb{Z}) \ $.

2.6.4) Define $R= \mathbb{C}[x,y]/\langle x^2 \rangle \ $.

Now, the Nullstellensatz gives us $\text{maxSpec}(\mathbb{C}[x,y])= \left\{{ \langle x-a_1,y-a_1 \rangle  : a_1, a_2\in\mathbb{C} }\right\} \ $.

The elements of this which