Show every non-zero submodule of $M$ contains a minimal submodule iff SocM is essential in M.

Question

I am studing the book "Rings and Categories of modules" written by Frank W. Anderson and Kent R. Fuller. On page 119, I am at a loss for the Corollary 9.10.

Corollary 9.10. Let $M$ be a left R-module. Then SocM is essential in M iff every non-zero submodule of $M$ contains a minimal submodule.

Let $M$ be a module. Then $Soc(M)=\sum\{N\leq M| \text{$N$ is a simple submodule of $M$}\}=\cap\{L\leq M| \text{$L$ is essential in $M$}\} $.

The writer says the Corollary 9.10. follws from the definition of SocM and the fact that $SocK=K\cap SocM$, where $K$ is any submodule of $M$.

I think I am an idiot，I still don't understand it. Any help will be appreciated.

Answer 1

You can get the corllary by the definitions of those terminologies:

$ \Rightarrow:$ If $SocM$ is essential in $M$, then for any non-zero submodule $N$ of $M$, $SocM \cap N \not =0$, that is $SocN \not =0$, so $N$ has a minimal submodule;

$\Leftarrow:$ If every non-zero submodule of $M$ contains a minimal submodule. Then the submodule must be simple(otherwise contradict with the minimality of the submodule), so every submodule $N \cap SocM \not =0$, then $SocM$ is essential in $M$.

Answer 2

I post my effort here. $SocM$ is essential in M iff $SocM\cap K=SocK=0$ implies $K=0$ for any submodule $K$ of $M$ iff $K\neq0$ implies $SocK\neq 0$ iff $K\neq0$, $K$ contains a minimal submodule.