Is a polynomial ring over a UFD in countably many variables a UFD? Let $R$ be a UFD. It is well know that $R[x]$ is also a UFD, and so then is $R[x_1,x_2,\cdots,x_n]$ is a UFD for any finite number of variables. Is $R[x_1,x_2,\cdots,x_n,\cdots]$ in countably many variables also a UFD? If not, what about if we take $\tilde{R} = R[x_1,\cdots,x_n,\cdots]$ where each polynomial must be given in finitely many of the variables?

Key Idea $ $ Each successive polynomial ring extension $\,D\subset D[x]\,$ is factorization inert, i.e. the ring extension introduces no new factorizations, i.e. if $\, 0\ne d\ \in D\,$ factors in $\,D[x]\,$ as $\,d = ab\,$ for $\, a,b\in D[x]\,$ then $\,a,b\in D.\,$ From this one easily deduces that the requisite factorization properties extend to the ascending union $\,R[x_1,x_2,\cdots\,].$  The same ideas works for arbitrary inert extensions.
Remark $\ $ Paul Cohn introduced the idea of inert extensions when studying Bezout rings. Cohn proved that every gcd domain can be inertly embedded in a Bezout domain, and every UFD can be inertly embedded in a PID. There are a few variations on the notion of  inertness that prove useful when studying the relationship between factorizations in base and extension rings, e.g. a weaker form where $\, d = ab\,\Rightarrow\, au, b/u\in D,\,$ for some unit $\,u\,$ in the extension ring.
