?-Supplemented modules and ?-weakly supplemented modules

Abstract

Given a hereditary torsion theory ? = (T, F) in Mod-R, a module M is called ?-supplemented if every submodule A of M contains a direct summand C of M with A/C ?-torsion. A submodule V of M is called ?-supplement of U in M if U + V = M and U ? V ? ?(V) and M is ?-weakly supplemented if every submodule of M has a ?-supplement in M. Let M be a ?-weakly supplemented module. Then M has a decomposition M = M<inf>1</inf> ? M <inf>2</inf> where M<inf>1</inf> is a semisimple module and M<inf>2</inf> is a module with ?(M<inf>2</inf>) ? M<inf>2</inf>. Also, it is shown that; any finite sum of ?-weakly supplemented modules is a ?-weakly supplemented module. © 2007 Elsevier B.V., All rights reserved.