I am working in the ring $R = \mathbb{Z}[\sqrt{−3}] = \{a + b\sqrt{−3} \; | \;a, b ∈ \mathbb{Z}\}$.
I am trying to find a maximal ideal $I$ of $R$ which properly contains $(2)$, and then prove that $I^{2}=(2).I$.
I was advised to begin by finding a homomorphism $\mathbb{R} → \mathbb{Z}/2\mathbb{Z}$, but don't know how to find this. I don't know if there is another way to find the ideal which I would find more intuitive?
Since $1^2 \equiv -3 \pmod{2}$, 1 behaves in $Z/2Z$ like a square root of -3, so you can define your homomorphism $f:R \to Z/2Z$ by $$ f(a + b \sqrt{-3}) = \overline{a+ 1 \cdot b} = \overline{a+ b} \quad a,b \in Z$$ where $\overline{a}$ denotes the class of the integer $a$ in $Z/2Z$.
I guess that the kernel of this homomorphism is the ideal that you are looking for! (It is indeed the only way of defining it! As $\overline{1}$ is the only one root of the polynomial $x^2+3$ in $Z/2Z$.