A simple module is necessarily the socle of its injective hull?

Question

According to the wikipedia page on injective hulls "a simple module is necessarily the socle of its injective hull". It is clear to me that why a simple module is a subset to the socle of of its injective hull, but I cannot prove the equality. Can anyone help me with that please?

Answer 1

An extension $M\hookrightarrow X$ is essential if any nonzero submodule of $X$ intersects $M$ nontrivially. If this is the case and if $M$ is moreover simple, then it follows that any simple submodule of $X$ equals $M$. Hence $$\text{soc}(X)=\sum\limits_{\substack{N\subset X\\\text{simple}}} N=M.$$ Note that the injectivity of $I$ is not important here.