Structure Sheaves and Noncommutative Topologies Original Research Article

In this note, we study abstract localization with respect to elements in the free semigroup generated by all Gabriel filters over a fixed base ring. The main purpose of this set-up, which behaves in many ways as the analogous one with respect to a single Gabriel filter, is to construct structure sheaves over noncommutative topologies associated to an arbitrary left noetherian ring. This construction is particularly useful, when the base ring is “very noncommutative” and does not allow for a sufficiently large spectrum of left or two-sided prime ideals.