w-Divisoriality in Polynomial Rings

TL;DR

This paper extends the characterization of divisorial domains to non-Noetherian cases and investigates whether polynomial extensions preserve the w-divisorial property, affirming this under certain conditions.

Contribution

It generalizes the Bass-Matlis characterization to non-Noetherian domains and establishes conditions under which polynomial extensions remain w-divisorial.

Findings

The characterization extends to non-Noetherian domains.

Polynomial rings over integrally closed w-divisorial domains are w-divisorial.

Polynomial rings over Mori w-divisorial domains are w-divisorial.

Abstract

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.