Polynomial extensions of IDF-domains and of IDPF-domains

Abstract

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.