Rings of Quotients

We are almost finished. Having shown how to construct the module of quotients Q_E(A) of any left R-module A at an equivalence class of injective modules, we are left only to show that Q_E(R) is in fact a ring and that Q_E(A) is, in a natural way, a left Q_E(R)-module for every left R-module A. We do this by a trick: namely we show that Q_E(R) is isomorphic, as a left R-module, to its own endomorphism ring R_E. Finally, we use the representation theory of the quotient module developed in Section §15.4 to give a characterization of this ring, namely we show that any injective left R-module E is equivalent to an injective left R-module E’ having endomorphism ring S such that Rg is isomorphic, as a ring, to Hom_s(E’,E’).