[2] For instance, taking S to be a full matrix ring over a field, and taking R to be any ring containing every matrix which is zero in all but the last column, the injective hull of the right R-module R is S. For instance, one can take R to be the ring of all upper triangular matrices.
[3] In particular, for an integral domain, the injective hull of the ring (considered as a module over itself) is the field of fractions.
This type of "ring of quotients" (as these more general "fields of fractions" are called) was pioneered in (Utumi 1956), and the connection to injective hulls was recognized in (Lambek 1963).
If C is locally small, satisfies Grothendieck's axiom AB5 and has enough injectives, then every object in C has an injective hull (these three conditions are satisfied by the category of modules over a ring).
[4] Every object in a Grothendieck category has an injective hull.