Boyle's Conjecture and perfect localizations

TL;DR

This paper investigates the properties of left QI-rings under perfect localizations and proves Boyle's conjecture for certain classes of these rings, expanding understanding of their structure and behavior.

Contribution

It demonstrates that perfect localizations preserve the left QI-ring property and confirms Boyle's conjecture for left QI-rings with finite Gabriel dimension and specific torsion theories.

Findings

Perfect localizations of left QI-rings remain left QI-rings.

Boyle's conjecture holds for left QI-rings with finite Gabriel dimension.

The conjecture is valid for rings satisfying the restricted left socle condition.

Abstract

In this article we study the behavior of left QI-rings under perfect localizations. We show that a perfect localization of a left QI-ring is a left QI-ring. We prove that Boyle's conjecture is true for left QI-rings with finite Gabriel dimension such that the hereditary torsion theory generated by semisimple modules is perfect. As corollary we get that Boyle's conjecture is true for left QI-rings which satisfy the restricted left socle condition, this result was proved first by C. Faith in \cite{faithhereditary}.