ABSTRACT
In this chapter we move from the notions of modules torsion and torsionfree with respect to an equivalence class of injective (or of projective) left R-modules to the construction of modules of quotients at such an equivalence class. The construction is done in two stages: beginning with a left R-module A, we first move via the natural map to a torsionfree factor module of A; then we embed this factor module in a certain submodule of its injective hull which is determined by a weak form of injectivity. What is surprising, and not at all clear at this stage, is that such a construction is, to use a term from category theory, “functorial”. Not only can we assign to each left R-module a module of quotients but also we can assign to each map between left R-modules a unique map between their modules of quotients.
