Let $R$ be the subring of all $z$ in $\mathbb{C}$ of the form

$$z=\frac{a+b \sqrt{-3}}{2}$$

where $a$ and $b$ are in $\mathbb{Z}$ and $a \equiv b(\bmod 2)$. Prove that $N(z)=z \bar{z}$ is a non-negative element of $\mathbb{Z}$, for all $z$ in $R$. Prove that the multiplicative group of units of $R$ has order 6 . Prove that $7 R$ is the intersection of two prime ideals of $R$.

[You may assume that $R$ is a unique factorization domain.]