slanglyn3u2
2022-04-25
Answered

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

Answered 2022-04-26
Author has **11** answers

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.

I write SM for the soccel of M. It is clear that

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