Severe right Ore sets and universal localisation

TL;DR

This paper introduces severe right Ore sets to analyze universal localizations of rings, providing new insights into module categories and simplifying classification of ring homomorphisms to division rings.

Contribution

It defines severe right Ore sets and demonstrates their use in describing finitely presented modules over universal localizations, offering a new proof of Cohn's classification.

Findings

Category of finitely presented modules over a universal localization is equivalent to a localization at a severe right Ore set

Provides a structural description of finitely presented modules over universal localizations

Simplifies the classification of homomorphisms from rings to division rings

Abstract

We introduce the notion of a severe right Ore set in the main as a tool to study universal localisations of rings but also to provide a short proof of P. M. Cohn's classification of homomorphisms from a ring to a division ring. We prove that the category of finitely presented modules over a universal localisation is equivalent to a localisation at a severe right Ore set of the category of finitely presented modules over the original ring. This allows us to describe the structure of finitely presented modules over the universal localisation as modules over the original ring.