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.

http://www.fc.up.pt/mp/clomp/

Date received: February 10, 1999