Definition: A non-zero module M is co-isosimple if it is isomorphic to all its nonzero quotients.
Definition: A proper submodule N of a non-zero module M is isomaximal if M/N is a co-isosimple module.
Definition: The left co-isoradical CI-rad(M) of a module M is the intersection of all isomaximal submodules of M.
Remark: Let A be the class of all co-isosimple modules. Then for each R-module M:
CI-rad(M)=$\cap${ker(p)|p : M $\rightarrow$ C, C $\in$ A and p an epimorphism}.
I just have these 3 new definitions and a remark to use there.
For the ($\Rightarrow$) part of the proof, I consider, let M be co-isosimple so it's simple by the hypothesis. Then we get:
CI-rad(M)=$\cap${ker(p)|p : M $\rightarrow$ S, S a simple module and p an epimorphism}, that is, it's the intersection of annihilators of simple modules and this is the definition of rad(M).
For this part I'm almost sure but the inverse implication I'm stuck. Is there anyone can help me? A little hint is enough for me..