It is shown, that a module B
is a rational extension of a submodule A if and only if B∕A is a torsion
module with respect to the largest torsion theory for which B is torsionfree.
The rational completion of a module can thus be viewed as a module of
quotients. The behavior of rationally complete modules under the formation
of direct sums and products is studied. It is also shown, that a module is
rationally complete provided it contains a copy of every nonprojective simple
module.
In the second part of the paper, rational extensions of modules over a left
perfect ring are studied. Necessary and sufficient conditions are given for a
semi-simple module to be rationally complete. This characterization depends only
on the idempotents of the ring. If R is left and right perfect and if every
simple right module is rationally complete, then every module is rationally
complete.
