Modules whose surjective endomorphisms have a \gamma-small kernels

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

Keywords:

Dedekind finite modules , Generalized Hopfian modules , gamma-Hopfian modules , Hopfian modules