The study of modules by properties of their endomorphisms has long been of interest. In this paper we introduce a proper generalization of that of Hopőan modules, called Jacobson Hopőan modules. A right R-module M is said to be Jacobson Hopőan, if any surjective endomorphism of M has a Jacobsonsmall kernel. We characterize the rings R for which every őnitely generated free R-module is Jacobson Hopőan. We prove that a ring R is semisimple if and only if every R-module is Jacobson Hopőan. Some other properties and characterizations of Jacobson Hopőan modules are also obtained with examples. Further, we prove that the Jacobson Hopőan property is preserved under Morita equivalences.
