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 automatically w-divisorial? We show that the answer is yes if D is either integrally closed or Mori.
GABELLI S, HOUSTON EVAN, & PICOZZA GIAMPAOLO (2009). w-Divisoriality in polynomial rings. COMMUNICATIONS IN ALGEBRA, 37, 1117-1127 [10.1080/00927870802278982].