Rank these notations from most valid to least valid $$\mathbb{Z}[\sqrt{-3}]$$
$$\mathbb{Z}\left[\frac{i \sqrt{3} - 1}{2}\right]$$
$${\bf{A}}(-3)$$
$$\mathbb{Z}[\omega]$$
$$\mathbb{Z}\left[-\frac{1}{2} + \frac{\sqrt{-3}}{2}\right]$$
$$\mathbb{Z}[\sqrt{-27}]$$
$$\mathbb{Z}\left[\frac{\sqrt{-3}}{2}\right]$$
I don't know how to number equations without a preamble, so please cut me a little slack.
Of course $\omega = -\frac{1}{2} + \frac{\sqrt{-3}}{2}$, right?
The notation with the bold A comes from a really old book found in some public libraries and used bookstores.

I would add $\mathcal{O}_{\mathbb{Q}(\sqrt{-3})}$ to the list and rank them thus:


*

*$\mathcal{O}_{\mathbb{Q}(\sqrt{-3})}$

*$\mathbb{Z}[\omega]$ (assuming $\omega$ has been defined)

*$\mathbb{Z}\left[-\frac{1}{2} + \frac{\sqrt{-3}}{2}\right]$

*$\mathbb{Z}\left[\frac{i \sqrt{3} - 1}{2}\right]$ (I wouldn't call myself an algebraic number theorist, but I don't like this one either, I guess I would prefer to limit $i$ to discussion of Gaussian integers).

*${\bf A}(-3)$ (Ethan Bolker used this one in a book that's available as a Dover reprint).


As others have already mentioned, neither $\mathbb{Z}[\sqrt{-3}]$ nor $\mathbb{Z}[\sqrt{-27}]$ are integrally closed.
Even though $\frac{\sqrt{-3}}{2} = \frac{1}{2} + \omega$, I don't know quite what to make of $\mathbb{Z}\left[\frac{\sqrt{-3}}{2}\right]$.
