Rational extensions of modules
Hans H. Storrer · Pacific Journal of Mathematics · 1971
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. 1* Filters and rational extensions* We first recall a number