Exact contexts, noncommutative tensor products, and universal localizations

Abstract

Exact contexts and their noncommutative tensor products are introduced which generalize the notions of Milnor squares and usual tensor products over commutative rings, respectively. Exact contexts are characterized by rigid morphisms which exist abundantly, while noncommutative tensor products not only capture some useful constructions in ring theory (such as coproducts of rings and trivially twisted extensions) but also provide a new method to construct universal localizations with rich homological and structural information. Moreover, sufficient and necessary conditions in terms of the data of exact contexts are presented to ensure that the universal localizations constructed are homological ring epimorphisms.