Vol. 38, No. 3, 1971

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Rational extensions of modules

Hans H. Storrer

Vol. 38 (1971), No. 3, 785–794

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.

Mathematical Subject Classification
Primary: 16A64
Received: 11 December 1970
Published: 1 September 1971
Hans H. Storrer