Noetherianity of a quadratic integer order

For a nonsquare integer $d$,The integers are Noetherian, the Hilbert basis theorem makes $\mathbb{Z}[X]$ Noetherian, and quotients preserve Noetherianity.

In $R=\mathbb{Z}[\sqrt{-5}]$, the idealhas additive index two. If $I=(a+b\sqrt{-5})$ were principal, multiplication by its generator would givewhich has no solution in integers. Thus $I$ is not principal.