A ring of quotients

Let $\Sigma$ be a radical filter in a ring $R$, and let the ring $Q$ be defined by the equation $Q=\mathrm{Hom}_H(E, E)$, where $H=\mathrm{Hom}_R(E, E)$ and $E$ is the $\Sigma$-envelope of the ring. We show that the ring $Q$ possesses the properties of a ring of quotients and coincides with the ring of quotients in the sense of Gabriel and Bourbaki if the annihilator of any ideal $I\in\Sigma$ is equal to zero.