# Socle of socle of module is the socle of the

Socle of socle of module is the socle of the module
$Soc\left(Soc\left(M\right)\right)=Soc\left(M\right)$
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Zemmiq34
Step 1
I write SM for the soccel of M. It is clear that $SSM\subset SM$. Now take $x\in SM$. The element x may be written as a finite linear combination $x={y}_{1}+\cdots +{y}_{n}$ where each ${y}_{r}$ is contained in a simple submodule ${V}_{r}$ of M. But each ${V}_{r}$ is also a simple submodule of SM, so in particular ${y}_{r}$ is contained in SSM. Since SSM is a submodule, it follows that x is contained in it.