Maximal Quotient Rings of Endomorphisms of Quasigenerators

O.Preliminaries. Let R be an associative ring with identity, and let Mod-R denote the category of all unital right R-modules. A set of right ideal of R is called a Gabriel topology on R if satisfies T1. If I∈ and I J, then J∈. T2. If I and J belong to, then I∩J∈. T3. I∈ and r∈R, then (I:r)={x∈R:rx∈I}∈. T4. If I is a right ideal of R and there exists J∈ such that (I:r)∈ for every r∈J, then I∈.