ABSTRACT
The paper is organized as follows: basic results and definitions are exposed as a starting point for the subsequent investigation in Section 2. In Section 3, some properties and motivating examples of divisorial multiplication rings are exposed. We characterize divisorial multiplication domains with respect to a Gabriel topology f. Krull domains turn out to be exactly Noetherian divisorial multiplication rings for the canonical Gabriel topology. In the last section we deal with Gabriel topologies T having enough .F-critical ideals. We have determined all Gabriel topologies over R such that R is a divisorial multiplication ring in the Noetherian case. Divisorial multiplication rings are characterized locally and new relative versions of Nakayama's Lemma and Mott's Theorem are shown. As a consequence, we can find many examples of divisorial multiplication rings. Finally, we determine the structure of Noetherian torsionfree divisorial multiplication ring relative to a Gabriel topology.
