Why is the generating set a proper ideal of...?

Question

Why is $\langle 89, 3-4\sqrt{-5}\rangle$ a proper ideal of $\Bbb{Z}[\sqrt{-5}]$?

Answer 1

A proper ideal is an ideal that is a strict subset of the ring (like proper subsets). So, that ideal is proper because (for example) $1$ is not an element of it. In fact, it is equivalent to being a proper ideal that the ideal does not contain $1$, since $1$ generates the whole ring.

Answer 2

As remarked by Wilf-Fine, $$89=(3-4\sqrt{-5})\cdot(3+4\sqrt{-5}) \in \langle 3-4\sqrt{-5}\rangle$$ and so $$\langle 89, 3-4\sqrt{-5}\rangle = \langle 3-4\sqrt{-5}\rangle$$

Now $$\alpha \in \langle 3-4\sqrt{-5}\rangle \implies N(\alpha) \in N(3-4\sqrt{-5})\mathbb Z = 89\mathbb Z$$ and so $1 \notin \langle 3-4\sqrt{-5}\rangle$ because $N(1)=1\notin 89\mathbb Z$.

Therefore, $\langle 3-4\sqrt{-5}\rangle$ is a proper ideal.