Is a ring $R$ factorial $\iff$ $R[X]$ factorial?

Question

Let $R$ be a factorial ring. Then, the polynomial ring $R[X]$ is factorial.

I was wondering if the other direction also works (i.e. $R[X]$ factorial $\implies$ $R$ factorial)?

If not, please give an example of a ring $R$ such that $R[X]$ is factorial, but $R$ is not.

Answer 1

• If $R[X]$ is integral, the $R$ is integral.
• The units in $R[X]$ are the units in $R$ (by lack of zero divisors).
• The irreducible elements of $R$ are the degree zero irreducible elements of $R[X]$.

Consequently, for $0\ne r\in R$, the unique factorization in $R[X]$ consists of degree zero factors and is a factorization in $R$, and is still unique.