Ideal generator of a quotient ring

Question

Assume J is a non-zero ideal of the ring $\mathbb{Z}[\alpha]= \{a+b\alpha:a,b \in \mathbb{Z} \}.$ I'm trying to prove that there exists a positive integer $n$ in J, but I don't know where to start.

Answer 1

I assume $\alpha \in \mathbb C$. Let $z = a + b\alpha$ be a non-zero element of an ideal $J$. Set $w = a - b\alpha$.

1. When $w = 0$. As $a = b\alpha$, we have $J \ni z = 2a \in \mathbb{Z} \setminus \{0\}$. So $z$ or $-z$ is what you're looking for.

2. When $w \neq 0$. Then $J \ni zw = a^2 - b^2\alpha^2 \in \mathbb{Z} \setminus \{0\}$. So $zw$ or $-zw$ does the job.

Added. If you're familiar with algebraic number theory, an approach suggested by Suzet and Alexandros would be better. This solution is too elementary that one might miss essence.