Cluesless over a proper ideal question

Question

Let $X=\{ \alpha+\beta\sqrt{-5} \mid \alpha,\beta \text{ are rational numbers} \}$

Does there exist an integer $a$ such that $a$ and $3-4\sqrt{-5}$ generate a proper ideal of $X$?

Can anyone answer this?

Answer 1

Observe that

$$\frac1{3-4\sqrt{-5}}=\frac{3+4\sqrt{-5}}{89}=\frac3{89}+\frac4{89}\sqrt{-5}\in X$$

So $\;3-4\sqrt{-5}\;$ is invertible and any ideal containing it is the whole $\;X\;$, which doesn't surprise since $\;X\;$ is in fact a field.