ABSTRACT
It is one of the fortunate basic facts of module theory that for each ring R, every R-module can be embedded in an injective R-module. Baer’s criterion indicates how we should begin the embedding process. If A is a left R-module which is not injective, then there is a left ideal L of R and a map α:L → A that cannot be enlarged to a map from R to A. It is an easy matter to embed A in a larger module B so that α can be enlarged to a map from R to B. This embedding process, although simple, is the most fundamental construction in this chapter, and is described rather formally.
OBSERVATION 1. For a left R-module A, a left ideal L of R, and a map α:L → A, the graph gr(−α) of the map −α is a submodule of R ∐ A. The function λ:Α → (R ∐ A)/gr(−α) defined by aλ = (0, a) + gr(−α) is an R-monomorphism. The function β:R → (R ∐ A)/gr(−α) defined by rβ = (r,0) + gr(−α) is a map that enlarges the composite αλ.
