Modules in which every surjective endomorphism has a \(\delta\)-small kernel

Abstract

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