Jacobson Hopfian modules

Abstract

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 Hopfian modules, called Jacobson Hopfian modules. A right \(R\)-module \(M\) is said to be Jacobson Hopfian, if any surjective endomorphism of \(M\) has a Jacobson-small kernel. We characterize the rings \(R\) for which every finitely generated free \(R\)-module is Jacobson Hopfian. We prove that a ring \(R\) is semisimple if and only if every \(R\)-module is Jacobson Hopfian. Some other properties and characterizations of Jacobson Hopfian modules are also obtained with examples. Further, we prove that the Jacobson Hopfian property is preserved under Morita equivalences.