Localizable sets and the localization of a ring at a localizable set

TL;DR

This paper introduces and studies localizable sets, generalizing Ore and denominator sets, and explores their properties and relationships in the context of semiprime Goldie rings.

Contribution

It defines localizable sets and demonstrates their equivalence with other localization concepts in semiprime Goldie rings, providing explicit descriptions.

Findings

Maximal left localizable sets containing all regular elements equal maximal left denominator sets.

In semiprime Goldie rings, five types of maximal sets coincide.

Explicit descriptions of these maximal sets are provided.

Abstract

The concepts of localizable set, localization of a ring and a module at a localizable set are introduced and studied. Localizable sets are generalization of Ore sets and denominator sets, and the localization of a ring/module at a localizable set is a generalization of localization of a ring/module at a denominator set. For a semiprime left Goldie ring, it is proven that the set of maximal left localizable sets that contain all regular elements is equal to the set of maximal left denominator sets (and they are explicitly described). For a semiprime Goldie ring, it is proven that the following five sets coincide: the maximal Ore sets, the maximal denominator sets, the maximal left or right or two-sided localizable sets that contain all regular elements (and they are explicitly described).