DECOMPOSITIONS OF MODULES SUPPLEMENTED RELATIVE TO A TORSION THEORY

Abstract

Let R be a ring, M a right R-module and a hereditary torsion theory in Mod-R with associated torsion functor τ for the ring R. Then M is called τ-supplemented when for every submodule N of M there exists a direct summand K of M such that K ≤ N and N/K is τ-torsion module. In [4], M is called almost τ-torsion if every proper submodule of M is τ-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′ ⊕ M″ be a direct sum of a semisimple module M′ and τ-supplemented module M″. Is M τ-supplemented? (2) Can one find a non-stable hereditary torsion theory τ and τ-supplemented modules M′ and M″ such that M′ ⊕ M″ is not τ-supplemented? (3) Can one find a stable hereditary torsion theory τ and a τ-supplemented module M such that M/N is not τ-supplemented for some submodule N of M? (4) Let τ be a non-stable hereditary torsion theory and the module M be a finite direct sum of almost τ-torsion submodules. Is M τ-supplemented? (5) Do you know an example of a torsion theory τ and a τ-supplemented module M with τ-torsion submodule τ(M) such that M/τ(M) is not semisimple?