The classical left regular left quotient ring of a ring and its semisimplicity criteria

TL;DR

This paper extends Goldie's semisimplicity criteria to the classical left regular left quotient ring, providing new conditions for its semisimplicity and deriving related criteria for the classical left quotient ring.

Contribution

It introduces semisimplicity criteria for the classical left regular left quotient ring, generalizing Goldie's theorem and offering new insights into ring semisimplicity conditions.

Findings

Semisimplicity criteria for the classical left regular left quotient ring.

Two new semisimplicity criteria for the classical left quotient ring.

Extension of Goldie's theorem to broader quotient rings.

Abstract

Let $R$ be a ring, $\CC_R$ and $\pCCR$ be the set of regular and left regular elements of $R$ ($\CC_R\subseteq \pCCR$). Goldie's Theorem is a semisimplicity criterion for the classical left quotient ring $Q_{l,cl}(R):=\CC_R^{-1}R$. Semisimplicity criteria are given for the classical left regular left quotient ring $'Q_{l,cl}(R):=\pCCR^{-1}R$. As a corollary, two new semisimplicity criteria for $Q_{l,cl}(R)$ are obtained (in the spirit of Goldie).