Socle of socle of module is the socle of the

slanglyn3u2 2022-04-25 Answered
Socle of socle of module is the socle of the module
Soc(Soc(M))=Soc(M)
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Answers (1)

Zemmiq34
Answered 2022-04-26 Author has 11 answers
Step 1
I write SM for the soccel of M. It is clear that SSMSM. Now take xSM. The element x may be written as a finite linear combination x=y1++yn where each yr is contained in a simple submodule Vr of M. But each Vr is also a simple submodule of SM, so in particular yr is contained in SSM. Since SSM is a submodule, it follows that x is contained in it.
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