Are Socle of a module and it's injective hull same?

Question

If $E(M)$ is an injective hull of an $R$-$module$ $M$ then $Soc(M)=Soc(E( M))$.

My attempt- abviously $Soc(M) \subset Soc(E(M))$ and since every essential submodule of $M$ is also an essential submodule of $E(M)$, $Soc(M) \supset Soc(E(M))$ (as socle of a module is the intersection of all its essential submodules).

Answer 1

Yes, that would be one way to do it:

$Soc(M)\subset Soc(E(M))$ since the simple submodules of $M$ are again simple submodules of $E(M)$.

And since you know the characterization of the socle as the intersection of essential submodules, and you know that essentiality is transitive, $Soc(M)\supseteq Soc(E(M))$.

It's even easier to reason directly that every simple submodule of $E(M)$ is a simple submodule of $M$. For if $S$ is a simple submodule of $E(M)$, we have immediately $M\supseteq S\cap M=S$ since $M\subseteq_e E(M)$.