Recall that a module $M$ is called isosimple if its each nonzero submodule is isomorphic to $M$. A module $M$ is called $J$-semisimple if $J(M)=0$, where $J(M)$ is the sum of all superfluous (small) submodules of $M$, or equivalently, intersection of all maximal submodules of $M$, and called as Jacobson radical of $M$.
I came across with the problem $``$Is every isosimple module $J$-semisimple?$"$ I tried to prove it but could not find any way. I also tried to construct counter examples to disprove it but could not find any such example. Kindly give any hint to prove or disprove it.
Have you tried a DVR as a module over itself? It should be isosimple by being a PID and not J-semisimple by being local with a non-zero maximal ideal.