Hopfcity and Jacobson small submodules

The study of modules by properties of their endomorphisms has long been of interest. In this paper, we introduce the notion of jacobson weakly Hopfian modules. It is shown that over a ring R, every projective (free) R-module is jacobson weakly Hopfian if and only if R has no nonzero semisimple projective R-module. Let L be a module such that L satisfies ascending chain conditions on jacobson-small submodules. Then it is shown that L is jacobson weakly Hopfian. Some basic characterizations of projective jacobson weakly Hopfian modules are proved.