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 $$c|a\and c|b \iff c|ab $$ 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