Ring Epimorphisms, Gabriel Topologies and Contramodules

Abstract

During the 1960s considerable work was done in order to understand the meaning of “epimorphism”. The notion plays an important role in categories of rings where the abstract category-theoretic meaning is now of common use.

The notion of ring epimorphism has relations with torsion theory and localisation theory. In particular, perfect right Gabriel topologies (in Stenström’s terminology) correspond bijectively to left flat ring epimorphisms.

In these notes we will consider two classes of modules defined in terms of a ring epimorphism: the comodules and the contramodules as introduced by Leonid Positselski. Adding mild conditions on the ring epimorphism we will extend classical results proved by Matlis for commutative rings by showing an equivalence between suitable subcategories of the two classes of comodules and contramodules.