Given any injective module I R , its bicommutator S contains as a subring the ring Q of quotients of R relative to the largest torsion theory for which I R is torsionfree. Q is even a dense subring of S if I R is sufficiently “nice”: every torsionfree factor module of Q R is divisible. Then S is the completion of Q in the finite topology, which here coincides with a certain I-adic topology, generalizing the usual P-adic topology on finitely generated modules over Noetherian local rings. Examples of nice injectives are: all injectives when R is right hereditary, injectives with zero singular submodule and the injective hull of R/ P when R is commutative and P is a prime ideal. It will be noticed that, in a number of concepts depending on the injective module I R , explicit reference to the latter has been suppressed. Thus we write “torsion” rather than “ I-torsion”, but we have retained “ I-adic” to contrast with “ P-adic”. The present paper supplements [8], but some care has been taken to make it self-contained.