We prove that if $R$ is an idempotent reflexive left Goldie ring whose simple singular left $R$-modules are GP-injective, then $R$ is a finite product of simple left Goldie rings. As a byproduct of this result we are able to show that if $R$ is semiprime, left Goldie and left weakly $\pi$-regular, then $R$ is a finite product of simple left Goldie rings.
Jin Yong Kim. "Certain rings whose simple singular modules are GP-injective." Proc. Japan Acad. Ser. A Math. Sci. 81 (7) 125 - 128, Sept. 2005. https://doi.org/10.3792/pjaa.81.125