  We extend the Bass-Matlis characterization of local Noetherian divisorial
domains to the non-Noetherian case. This result is then used to study the
following question: If a domain D is w-divisorial, that is, if each w-ideal of
D is divisorial, then is D[X] automatically w-divisorial? We show that the
answer is yes if D is either integrally closed or Mori.
