Boyle's Conjecture and perfect localizations

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}.