On H-Supplemented Modules

Abstract

A module M is called H-supplemented if for every submodule A of M there is a direct summand A′ of M such that A + X = M holds if and only if A′ +X = M for each submodule X of M. The eighteenth question (Is every H-supplemented module amply supplemented? p. 106) posed by Mohamed and Müller in 1990 is answered in the negative. It is well known that the direct sum of H-supplemented modules is not in general H-supplemented. Some sufficient conditions such that the direct sum of H-supplemented modules is H-supplemented are given.

Key Words:

• H-supplemented module
• Lifting module
• Sjective module

2000 Mathematics Subject Classification:

• 16D10
• 16D50

ACKNOWLEDGMENTS

The authors would like to thank the referee for his or her valuable comments. The second author was partially supported by NSF of China (Grant No. 11001222).

Notes

Communicated by T. Albu.