BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, vol.37, no.3, pp.703-717, 2014 (SCI-Expanded)
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 delta-supplemented modules and principally circle plus-supplemented modules. The module M is called principally circle plus-delta-supplemented if for any m is an element of M there exists a direct summand A of M such that M = mR + A and mR boolean AND A is delta-small in A. We prove that some results of principally circle plus-supplemented modules can be extended to principally circle plus-delta-supplemented modules for this general setting. Several properties of these modules are given and it is shown that the class of principally circle plus-delta-supplemented modules lies strictly between classes of principally circle plus-supplemented modules and principally delta-supplemented modules. We investigate conditions which ensure that any factor modules, direct summands and direct sums of principally circle plus-delta-supplemented modules are also principally circle plus-delta-supplemented. We give a characterization of principally circle plus-delta-supplemented modules over a semisimple ring and a new characterization of principally delta-semiperfect rings is obtained by using principally circle plus-delta-supplemented modules.