On a class of ?-supplemented modules

Let R be an arbitrary ring with identity and M a right R-module. In this paper, we introduce a class of modules which is an analogous to that of ?-supplemented modules and principally ?-supplemented modules. The module M is called principally ?-?- supplemented if for any m ? M there exists a direct summand A of M such that M = mR+A and mR ? A is ?-small in A. We prove that some results of principally ?-supplemented modules can be extended to principally ?-?-supplemented modules for this general setting. Several properties of these modules are given and it is shown that the class of principally ?- ?-supplemented modules lies strictly between classes of principally ?-supplemented modules and principally ?-supplemented modules. We investigate conditions which ensure that any factor modules, direct summands and direct sums of principally ?-?-supplemented modules are also principally ?-?-supplemented. We give a characterization of principally ?-?-supplemented modules over a semisimple ring and a new characterization of principally ?-semiperfect rings is obtained by using principally ?-?-supplemented modules.