There is a simple criterion using norms to test if such modules are ideals. If $M = [a,b+\sqrt{d}]$ is an ideal then it contains the norm $ N(b+\sqrt{d}) = (b+\sqrt{d})(b-\sqrt{d}) = b^2-d,$ so $a|b^2-d.$ This necessary condition is also sufficient. The proof is easy:
The module $M = [a,b+\sqrt{d}]$ is an ideal of $R = \Bbb Z[\sqrt{d}]⟺M$ is closed under multiplication by elements of $R⟺\sqrt{d}M⊆M⟺a\sqrt{d}, (b+\sqrt{d})\sqrt{d} \in M.$ The first inclusion is clear since $a\sqrt{d} = a(b+\sqrt{d})-ab\in M.$ For the second inclusion\begin{eqnarray*}-(b+\sqrt{d})\sqrt{d} &=& (b+\sqrt{d})(b-\sqrt{d})-b(b+\sqrt{d})\\ &=& b^2-d -b(b+\sqrt{d})\end{eqnarray*} The prior is $∈M = [a,b+\sqrt{d}]⟺a|b^2-d = N(b+\sqrt{d})$ where $N = $ the norm.
This is true for $[2,1+\sqrt{-5}],$ i.e. $2|N(1+\sqrt{-5}) = (1+\sqrt{-5})(1-\sqrt{-5}) = 1+5 = 6.$
The criterion generalizes to test idealness of the module $[a,b+cω]$ in the ring of integers of a quadratic number field, e.g. see section 2.3 in Franz Lemmermeyer's notes.
This is a special case of module normal forms that generalize to higher degree number fields, e.g. see the discussion on Hermite and Smith normal forms in Henri Cohen's A Course in Computational Number Theory.
The module $M = [a,b+\sqrt{d}]$ is an ideal of $R = \Bbb Z[\sqrt{d}]⟺M$ is closed under multiplication by elements of $R⟺\sqrt{d}M⊆M⟺a\sqrt{d}, (b+\sqrt{d})\sqrt{d} \in M.$ The first inclusion is clear since $a\sqrt{d} = a(b+\sqrt{d})-ab\in M.$ For the second inclusion\begin{eqnarray*}-(b+\sqrt{d})\sqrt{d} &=& (b+\sqrt{d})(b-\sqrt{d})-b(b+\sqrt{d})\\ &=& b^2-d -b(b+\sqrt{d})\end{eqnarray*} The prior is $∈M = [a,b+\sqrt{d}]⟺a|b^2-d = N(b+\sqrt{d})$ where $N = $ the norm.
This is true for $[2,1+\sqrt{-5}],$ i.e. $2|N(1+\sqrt{-5}) = (1+\sqrt{-5})(1-\sqrt{-5}) = 1+5 = 6.$
The criterion generalizes to test idealness of the module $[a,b+cω]$ in the ring of integers of a quadratic number field, e.g. see section 2.3 in Franz Lemmermeyer's notes.
This is a special case of module normal forms that generalize to higher degree number fields, e.g. see the discussion on Hermite and Smith normal forms in Henri Cohen's A Course in Computational Number Theory.