Find a nontrivial proper ideal of $\mathbb{Z}\times\mathbb{Z}$ that is not prime

Question

So I know that $4\mathbb{Z}\times\mathbb{Z}$ is a non-prime ideal of $\mathbb{Z}\times\mathbb{Z}$, and why it is. My question is, how would you find this without testing out many different ideals of $\mathbb{Z}\times\mathbb{Z}$?

Answer 1

It's pretty clear that $a = (2, 0) \not \in 4\mathbb{Z}\times\mathbb{Z}$, while $a^2 = (4, 0) \in 4\mathbb{Z}\times\mathbb{Z}$.

There's a general method of finding whether given ideal of $I \subset \mathbb{Z}^n$ is prime. Let $I = (a_1, \ldots, a_k)$. Then, $I$ is an image of an obvious map $\mathbb{Z}^k \to \mathbb{Z^n}$. This map is given by a matrix with $a_i$ as columns. Now, by a process quite similar to Gaussian elimination, we can transform this matrix to a diagonal matrix, the image of which in $\mathbb{Z^n}$ will also be $I$ -- this is just finding another set of generators of $I$. However, with diagonal generators it is now clear whether the ideal is prime -- in your problem above, the diagonal generators are $(4, 0), (0, 1)$, and it's pretty clear $(2, 0)$ is not generated by them, while $(2, 0)^2$ belongs to the ideal.

The procedure I mentioned is basically a classification of finitely generated modules over a PID, so look for how this classification works.