Is $\mathbb{Z}\Bigl[\frac{1}{2},\frac{1}{3}\Bigr]$ a Dedekind domain? Is $\mathbb{Z}\Bigl[\frac{1}{2},\frac{1}{3}\Bigr]$ a Dedekind Domain? Can anyone help me with a detailed reasoning?

This answer is distilled from Gone's and rola's comments:

By this question, and since $\Bbb Z \subset \Bbb Z[\frac12,\frac13] \subset \Bbb Q = Q(\Bbb Z)$, we conclude $\Bbb Z[\frac12,\frac13]$ is a PID. PIDs are Dedekind, so we are done.
(More generally, any localization of a PID is again a PID.)

