Polynomial extensions of IDF-domains and of IDPF-domains

An integral domain is IDF if every non-zero element has only finitely many non-associate irreducible divisors. We investigate when $R$ IDF implies that the ring of polynomials $R[T]$ is IDF. This is true when $R$ is Noetherian and integrally closed, in particular when $R$ is the coordinate ring of a non-singular variety. Some coordinate rings $R$ of singular varieties also give $R[T]$ IDF. Analogous results for the related concept of IDPF are also given. The main result on IDF in this paper states that every countable domain embeds in another countable domain $R$ such that $R$ has no irreducible elements, hence vacuously IDF, and the polynomial ring $R[T]$ is not IDF. This resolves an open question. It is also shown that some subrings $R$ of the ring of Gaussian integers known to be IDPF also have the property that $R[T]$ is not IDPF.