Let $I=$($2,1＋\sqrt{-5}$)$\in \Bbb{Z}[ \sqrt{-5}]$. $I$ is an ideal of $\Bbb{Z}[ \sqrt{-5}]$, so it must be free $\Bbb{Z}$module of rank $2$.
But I'm having trouble to recognize the structure as $\Bbb{Z}$module. If we write $I \cong A\Bbb{Z}＋B\Bbb{Z}$,
What can we take as $A$ and $B$ ? (In other words, $I$ forms the lattice of $ \Bbb{C}$, but how the lattice looks like?)
My try: $2(a＋b > \sqrt{-5})＋(1＋\sqrt{-5})(c＋d\sqrt{-5})=(2a＋c-5d)＋(2b＋d＋c)\sqrt{-5}=2(a-b-3d)＋(２b＋d＋c)(1＋\sqrt{-5})$,
thus $2$ and $1＋\sqrt{-5} $ spans $I$ with integer coefficient, and obviously they are linearly independent.
This is why I believe $A=2,B= 1＋\sqrt{-5} $ satisfies the condition, but I don't have confident. Is my try correct ?
Thank you in advance.