Modules That Have a Delta-Supplement in Every Extension

Abstract

Let R be a ring and M be a left R-module. In this paper, we define modules with the properties (delta-E) and (delta-EE), which are generalized version of Zoschinger's modules with the properties (E) and (EE), and provide various properties of these modules. We prove that the class of modules with the property (6-E) is closed under direct summands and finite direct sums. It is shown that a module M has the property (delta-EE) if and only if every submodule of M has the property (delta-E). It is a known fact that a ring R is perfect if and only if every left R-module has the property (E). As a generalization of this, we prove that if R is a delta-perfect ring then every left R-module has the property (delta-E). Moreover, the converse is also true on delta-semiperfect