On H-Supplemented Modules

Abstract

Let [Formula: see text]be a finite direct sum of modules. We prove: (i) If M i is radical M j -projective for all j > i and each M i is H-supplemented, then M is H-supplemented. (ii) If all the M i are relatively projective and N is H-supplemented, then each M i is H-supplemented. Let ρ be the preradical for a cohereditary torsion theory. Let M be a module such that ρ (M) has a unique coclosure and every direct summand of ρ (M) has a coclosure in M. Then M is H-supplemented if and only if there exists a decomposition M=M 1 ⊕M 2 such that M 2 ⊆ ρ(M), ρ(M)/M 2 ≪ M/M 2 , and M 1 , M 2 are H-supplemented.