Modules in which every surjective endomorphism has a δ-small kernel

Abstract

Hopfian modules, δ-Hopfian modules. In this paper, we introduce the notion of δ-Hopfian modules. We give some properties of these modules and provide a characterization of semisimple rings in terms of δ-Hopfian modules by proving that a ring R is semisimple if and only if every R-module is δ-Hopfian. Also, we show that for a ring R, δ(R) = J(R) if and only if for all R-modules, the conditions δ-Hopfian and generalized Hopfian are equivalent. Moreover, we prove that δ-Hopfian property is a Morita invariant. Further, the δ-Hopficity of modules over truncated polynomial and triangular matrix rings are considered.