Let $M$ be an $A$-module where $A$ is a $k$-algebra ($k$ an algebraically closed field). If $\operatorname{Soc}{(M)}$ denotes the socle of $M$, i.e the sum of all simple submodules of $M$ why is that that if $N$ is a nonzero submodule of $M$ then $N$ has non trivial intersection with $\operatorname{Soc}{(M)}$?
Sum of all simple submodules
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representation-theory
socle
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0No this still won't do; The module $Soc(\mathbb{\overline{Q}}[X]) = 0$ where $\mathbb{\overline{Q}}[X]$ is considered as a $\mathbb{\overline{Q}}[X]$-module. – 2011-09-12
1 Answers
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Prove that the socle of every non-zero finite dimensional module is non-zero. In other words, that every non-zero finite dimensional module contains a simple one.
Can you see how to use this to conclude?