Essentially iso-retractable modules and rings

A.K. Chaturvedi et al. (2021) call a module $M$ essentially iso-retractable if for every essential submodule $N$ of $M$ there exists an isomorphism $f : M\rightarrow N.$ We characterize essentially iso-retractable modules, co-semisimple modules ($V$-rings), principal right ideal domains, simple modules and semisimple modules. Over a Noetherian ring, we prove that every essentially iso-retractable module is isomorphic to a direct sum of uniform submodules.