In this proof of "$R$ is UFD $\Rightarrow$$R[x]$ is UFD", the author showed that any nonzero $f \in R[x]$ can be factorized into primes. Then how does $R[x]$'s UFD property follow?
I believe we also need:
If $R$ is integral domain: $R$ is UFD $\Leftrightarrow$ every irreducible is prime, every non unit factors into irreducible $\Leftrightarrow$every non unit factors into primes.
Or is the above unnecessary?