Modules that Have a δ-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 (δ-E) and (δ-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 (δ-E) is closed under direct summands and finite direct sums.It is shown that a module M has the property (δ-EE) if and only if every submodule of M has the property (δ-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 also prove that a ring R is δ-perfect if and only if every left R-module has the property (δ-E).