Let I be an ideal generated by $\displaystyle 2$ and $\displaystyle 1+\sqrt{-3}$ in the ring $\displaystyle \mathbb Z[{\sqrt{-3}}].$

Show that $\displaystyle I$ is different $\displaystyle (2).$

And $\displaystyle I^2 =2I.$

Show that ideals in $\displaystyle \mathbb Z[{\sqrt{-3}}].$ do not factor uniquely into prime ideals.

Show that I is a unique prime ideal contaning $\displaystyle (2).$

Conclude $\displaystyle (2)$ is not a product of prime ideals.