Weakly left localizable rings

TL;DR

This paper introduces the class of weakly left localizable rings, characterizes them via criteria involving maximal left denominator sets, and describes their structure as products of local rings with nil radicals.

Contribution

It defines weakly left localizable rings and provides explicit criteria and structural descriptions for rings with finitely many maximal left denominator sets.

Findings

Weakly left localizable rings are characterized by their maximal left denominator sets.

Such rings with finitely many maximal left denominator sets have a left quotient ring as a product of local rings.

The radicals of these local rings are nil ideals.

Abstract

A new class of rings, {\em the class of weakly left localizable rings}, is introduced. A ring $R$ is called {\em weakly left localizable} if each non-nilpotent element of $R$ is invertible in some left localization $S^{-1}R$ of the ring $R$. Explicit criteria are given for a ring to be a weakly left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case if a ring has a left Artinian left quotient ring). It is proved that a ring with finitely many maximal left denominator sets that satisfies some natural conditions is a weakly left localizable ring iff its left quotient ring is a direct product of finitely many local rings such that their radicals are nil ideals.