The largest strong left quotient ring of a ring

TL;DR

This paper introduces and studies the largest strong left quotient ring of a ring, exploring its properties, relationships with classical quotient rings, and explicit descriptions for various classes of rings.

Contribution

It defines the largest strong left quotient ring and the strong left localization radical, providing properties, criteria, and explicit descriptions for different classes of rings.

Findings

$Q_l^s(Q_l^s(R))$ is isomorphic to $Q_l^s(R)$

The strong left localization radical $ ext{gll}_R^s$ is zero for certain rings

A criterion for $Q_l^s(R)$ to be semisimple

Abstract

For an arbitrary ring $R$, the largest strong left quotient ring $Q_l^s(R)$ of $R$ and the strong left localization radical $\glsR$ are introduced and their properties are studied in detail. In particular, it is proved that $Q_l^s(Q_l^s(R))\simeq Q_l^s(R)$, $\gll^s_{R/\glsR}=0$ and a criterion is given for the ring $ Q_l^s(R)$ to be a semisimple ring. There is a canonical homomorphism from the classical left quotient ring $Q_{l, cl}(R)$ to $Q_l^s(R)$ which is not an isomorphism, in general. The objects $Q_l^s(R)$ and $\gll^s_R$ are explicitly described for several large classes of rings (semiprime left Goldie ring, left Artinian rings, rings with left Artinian left quotient ring, etc).