I want to check if $R = (\mathbb{Z}/2\mathbb{Z})[T,T^{-1}]$ is a DVR.
What I tried is showing that/ checking if $R$ is a local noetherian integrally closed domain, with precisely two prime ideals, the zero ideal and the maximal ideal. However, I already get stuck at the integrally closed part.
How should I do this, resp. is there an easier way to check if $R$ is a DVR?
The ring $R = (\mathbb{Z}/2\mathbb{Z})[T]$ is a polynomial ring over a field, so it's a Principal Ideal Domain.
The ring $B = (\mathbb{Z}/2\mathbb{Z})[T,T^{-1}]$ (known as the ring of Laurent polynomials over $\mathbb{Z}/2\mathbb{Z}$), is the localization of $R$ at the multiplicative set generated by $T$. Localizations of PIDs are still PIDs, so $B$ is a PID. In particular $B$ is locally a DVR. But $B$ has infinitely many prime ideals, so it's far from local, and certainly not a DVR. To be precise, the prime ideals of $B$ are in one-to-one correspondence with the prime ideals of $R$ that don't contain $T$. The only prime ideal of $R$ that contains $T$ is $(T)$.