Modules whose surjective endomorphisms have ay-small kernels

In this paper, we introduce a proper generalization of that of Hopfian modules, called -Hopfian modules. A right -module is said to be -Hopfian, if any surjective endomorphism of has a -small kernel. Some basic characterizations of -Hopfian modules are proved. We prove that a ring is semisimple cosingular if and only if every -module is -Hopfian. Further, we prove that the -Hopfian property is preserved under Morita equivalences. Some other properties of -Hopfian modules are also obtained with examples.