I wonder if either the coordinate ring $A(X):=\mathbb{C}[x,y]/(x^2+y^2-1)$ or the polynomial ring $\mathbb{C}[t,t^{-1}]$ are unique factorization domains?
I know that those are isomorphic, so the answere for one of those would do it. In my trys of solving this issue, I saw that: $y^2 = (1-x)(1+x)$ ist a none-unique factorization in irreducible elements, because $y, (1-x)$ and $(1+x)$ are irreducible in $\mathbb{C}[x,y]/(x^2+y^2-1)$. But I can't even prove, that $y$ is irreducible. I know, that there are no none-units nor units, which I could multiply and get $y$ again. As far as I know, the units in $A(X)$ are the constant polynomials.
I hope that someone can understand my issue (so in advance in hope you can read my spelling w/o getting hurt :) ..) and make this topic clear to me.
Thanks
Observe that $\mathbb{C}[t,t^{-1}]=\mathbb{C}[t]_t$. Now, as localization of a UFD is a UFD, we are done.
In fact, we actually have a stronger result. Localization of a PID is again a PID . Since $\mathbb{C}[t]$ is a PID, $\mathbb{C}[t,t^{-1}]$ is a PID as well.