Expert Answer to "Socle of socle of module is the socle..."

slanglyn3u2

Answered question

2022-04-25

Socle of socle of module is the socle of the module

S o c (S o c (M)) = S o c (M)

Zemmiq34

Beginner2022-04-26Added 11 answers

Step 1
I write SM for the soccel of M. It is clear that

S S M \subset S M

. Now take

x \in S M

. The element x may be written as a finite linear combination

x = y_{1} + \dots + 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.

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