The largest left quotient ring of a ring

TL;DR

This paper introduces the largest left quotient ring of a ring, proves its properties, and explores its relation to classical quotient rings, extending Ore's localization method with applications to various ring classes.

Contribution

It establishes the existence of the largest left quotient ring and extends Ore's localization method to localizable left Ore sets.

Findings

The largest left quotient ring $Q_l(R)$ exists for any ring.

$Q_l(R)$ is semi-simple iff the classical quotient ring exists and is semi-simple.

If $Q_l(R)$ is left artinian, then it coincides with the classical quotient ring.

Abstract

The left quotient ring (i.e. the left classical ring of fractions) $Q_{cl}(R)$ of a ring $R$ does not always exist and still, in general, there is no good understanding of the reason why this happens. In this paper, it is proved existence of the largest left quotient ring $Q_l(R)$, i.e. $Q_l(R) = S_0(R)^{-1}R$ where $S_0(R)$ is the largest left regular denominator set of $R$. It is proved that $Q_l(Q_l(R))=Q_l(R)$; the ring $Q_l(R)$ is semi-simple iff $Q_{cl}(R)$ exists and is semi-simple; moreover, if the ring $Q_l(R)$ is left artinian then $Q_{cl}(R)$ exists and $Q_l(R) = Q_{cl}(R)$. The group of units $Q_l(R)^*$ of $Q_l(R)$ is equal to the set $\{s^{-1} t\, | \, s,t\in S_0(R)\}$ and $S_0(R) = R\cap Q_l(R)^*$. If there exists a finitely generated flat left $R$-module which is not projective then $Q_l(R)$ is not a semi-simple ring. We extend slightly Ore's method of localization to localizable left Ore sets, give a criterion of when a left Ore set is localizable, and prove that all left and right Ore sets of an arbitrary ring are localizable (not just denominator sets as in Ore's method of localization). Applications are given for certain classes of rings (semi-prime Goldie rings, Noetherian commutative rings, the algebra of polynomial integro-differential operators).