Let $M$ be an $A$-module. How do I show that Soc$(M)$ is the intersection $Q$ of all essential submodules of $M$? One direction is easy enough (Soc$(M)\subset Q$), but I can't seem to show the other direction. Only if I could show that every submodule of $Q$ is a direct summand of $Q$... Does anyone have any idea?
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This is e.g. proven in Proposition 7.19 of the representation theory lecture notes of Ringel and Schröer (here essential submodules are called large submodules). Here is a link http://www.math.uni-bonn.de/people/schroer/dst/dst_2009.pdf