Certain rings whose simple singular modules are GP-injective

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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.