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On QF rings and small modules.
by
Christian Lomp
Departamento de Matemática Pura, Faculdade de Ciências, Universidade do Porto, Portugal
Let R be an associative ring with unit. A left R-module is called small if it is a small submodule of its injective hull (e.g. the integers Z as Z-module). Özcan and Harmanci showed that every left R-module over a QF ring R can be decomposed into a direct sum of an X- and an X*-module, where X denotes the class of modules that does not contain any small left R-module and X* denotes the class of modules such that every subfactor contains a small left R-module. They raised the question if any ring with that property is already QF. In this talk we first realize (X*, X) as a hereditary torsion theory G* and then discuss when G* is splitting, i.e. every left R-module is a direct sum of a G*-torsion and a G*-torsionfree module. We show that this happens for the following classes of rings: left V-rings, local rings, commutative semiperfect rings, semilocal left Kasch rings and direct products of commutative proper integral domains. In particular we show that for any semilocal ring R: G* is (left) splitting if and only if R cogenerates all injective simple left R-modules.
Date received: February 10, 1999
Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-72.