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

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.