H-supplemented modules and singularity

Let M be a module over a ring R. We call M,\delta-H-supplemented provided for every submodule N of M there is a direct summand D of M such that M=N+X if and only if M=D+X for every submodule X of M with M/X singular. We prove that M is \delta-H-supplemented if and only if for every submodule N of M there exists a direct summand D of M such that (N+D)/N\ll_{\delta} M/N and (N+D)/D\ll_{\delta} M/D.

Keywords:

delta-small submodule , H-supplemented module , delta-H-supplemented module