Over a left V-ring, why is every module containing a simple essential submodule is simple?

Question

I've tried to prove the statement $R$ is a $V-$ring if and only if every module containing a simple essential submodule is simple. I know that a ring $R$ is called as a $V-$ring if every simple $R$ module is injective. However, I cannot prove the above statement by just using this definition of $V-$ring. Is there anyone can give me a hint for this, at least an extra information about injectivity or $V-$ring that I can use for the proof of that statement?

Answer 1

"$R$ is a V-ring if and only if every module containing a simple submodule is simple." is a false theorem.

For example, the ring $R=F_2\times F_2$ is a semisimple ring, and so all of its modules contain simple modules. Semisimple rings are also V-rings, but obviously not all of its modules can be simple.

To address the new version of the question

"$R$ is a V-ring if and only if every module containing an essential simple submodule is simple."

This is obvious since injective modules are summands of their supermodules. No summand can be essential unless it is already the entire module. So if a module contains an essential injective submodule, that submodule is already the entire module.