How do I prove that the socle of an Artinian module is an essential submodule?
I don't see where we should use the artinianity of the module to prove this.
2026-02-23 07:38:40.1771832320
Artinian module's socle is essential
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The artinian condition comes in when looking for a minimal submodule. Let $H \neq 0$ be a submodule such that $$H \cap {Soc}(N) = 0.$$ Then we wish to find a minimal submodule of $N$ contained in $H$. This is given by the artinian condition on $N$. But this minimal submodule should be contained in $Soc(N)$, by definition. Contradiction.