Some answers may be obvious but for the sake of symmetry let me please ask my question in this general form:
Consider the pairs of rings
$\mathbb{Z}[\frac{1}{2}]$ and $\mathbb{Z}[\frac{1}{3}]$
$\mathbb{Z}[\sqrt{2}]$ and $\mathbb{Z}[\sqrt{3}]$
$\mathbb{Z}[\sqrt{2}]$ and $\mathbb{Z}[\sqrt[3]{2}]$
Note that
$\mathbb{Z}[\frac{1}{2}] = \{a_0 + \frac{a_1}{2} + \dots + \frac{a_n}{2^n} \ |\ n \in \mathbb{N}, a_i \in \mathbb{Z}\}$
$\mathbb{Z}[\sqrt{2}] = \{a_0 + a_1\sqrt{2} + \dots + a_n\sqrt{2}^n \ |\ n \in \mathbb{N}, a_i \in \mathbb{Z}\} = \{a_0 + a_1\sqrt{2} \ |\ a_i \in \mathbb{Z}\}$
My question is:
Which ring properties do the members of each pair have in common and which ones not?
For $\mathbb{Z}[\frac{1}{2}]$ and $\mathbb{Z}[\frac{1}{3}]$ I cannot see which ring properties they could not have in common.
For $\mathbb{Z}[\sqrt{2}]$ and $\mathbb{Z}[\sqrt[3]{2}]$ I believe to know that there are such properties, due to the fact, that both square roots of $2$ are real while only one cubic root of $2$ is real.
For $\mathbb{Z}[\sqrt{2}]$ and $\mathbb{Z}[\sqrt{3}]$ I am not sure. I suspect they may have different properties, since Franz Lemmermeyer in his book Quadratische Zahlkörper (unfortunately only in German) discusses in detail the ring $\mathbb{Z}[\sqrt{3}]$ but not $\mathbb{Z}[\sqrt{2}]$ (which would have been the more natural example when both were essentially the same).
$\mathbb{Z}[\sqrt{2}]$, $\mathbb{Z}[\sqrt{3}]$, $\mathbb{Z}[\sqrt[3]{2}]$ are all Dedekind domains. They are even all Euclidean domains and so PIDs.
$\mathbb{Z}[\frac{1}{2}]$ and $\mathbb{Z}[\frac{1}{3}]$ are PIDs. So is every subring of $\mathbb{Q}$.
$\mathbb{Z}[\sqrt{5}]$ is not a Dedekind domain because it is not integrally closed. In fact, $\mathbb{Z}[\sqrt{d}]$ with $d$ squarefree is a Dedekind domain iff $d \equiv 2,3 \bmod 4$.