On the Splitting of the Dual Goldie Torsion Theory

TL;DR

This paper investigates the dual Goldie torsion theory, its splitting properties, and explores whether its splitting implies the ring is quasi-Frobenius, providing new insights into torsion theories in ring theory.

Contribution

It introduces an appropriate dual Goldie torsion theory, analyzes its splitting behavior, and answers a specific open question about the implications for ring structure.

Findings

The dual Goldie torsion theory can be split under certain conditions.

Splitting of the dual Goldie torsion theory does not necessarily imply the ring is quasi-Frobenius.

The paper clarifies the relationship between torsion theory splitting and ring properties.

Abstract

The splitting of the Goldie (or singular) torsion theory has been extensively studied. Here we determine an appropriate dual Goldie torsion theory, discuss its splitting and answer in the negative a question proposed by Ozcan and Harmanci as to whether the splitting of the dual Goldie torsion theory implies the ring to be quasi-Frobenius.