INTERNATIONAL JOURNAL OF MATHEMATICS, vol.16, no.1, pp.43-52, 2005 (Journal Indexed in SCI)
Let R be a ring, M a right R-module and a hereditary torsion theory in Mod-R with associated torsion functor tau for the ring R. Then M is called tau-supplemented when for every submodule N of M there exists a direct summand K of M such that K less than or equal to N and N/K is tau-torsion module. In [4], M is called almost tau-torsion if every proper submodule of M is tau-torsion. We present here some properties of these classes of modules and look for answers to the following questions posed by the referee of the paper [4]: (1) Let a module M = M' circle plus M" be a direct sum of a semisimple module M' and tau-supplemented module M". Is M tau-supplemented? (2) Can one find a non-stable hereditary torsion theory tau and tau-supplemented modules M' and M" such that M' circle plus M" is not tau-supplemented? (3) Can one find a stable hereditary torsion theory tau and a tau-supplemented module M such that M/N is not tau-supplemented for some submodule N of M? (4) Let tau be a non-stable hereditary torsion theory and the module M be a finite direct sum of almost tau-torsion submodules. Is M tau-supplemented? (5) Do you know an example of a torsion theory tau and a tau-supplemented module M with tau-torsion submodule tau(M) such that M/tau(M) is not semisimple?