Axiomatic definition of a quotient ring

The quotient ring $Q\Phi(R)$ of the ring $R$ with respect to a strong filter $\Phi$ of right ideals of $R$ is axiomatically defined as a maximal strong $\Phi$-essential extension of $R$. The ring $Q\Phi(R)$ is constructively obtained in the form of $\lim\operatorname{Hom_r}(I,R)$, where $I\in\Phi$.