Modules in which every surjective endomorphism has a $$\mu $$-small kernel

Jacobson Hopfian modules

“…In [5,Theorem 3], it is shown that a ring R is semisimple if and only if every R-module is δ-Hopőan. In [6], El Moussaouy and Ziane introduced and studied the concept of µ-Hopőan modules. A right R-module M is called µ-Hopőan if any surjective endomorphism of M has a µ-small kernel.…”

“…In [7], Ghorbani and Haghany proved that if ACC holds on nonsmall submodules of M , then M is generalized Hopőan. In [6], El Moussaouy and Ziane proved that if ACC holds on non µ-small submodules of M , then M is µ-Hopőan. Also we know that Noetherian modules are Hopőan modules.…”