# Socle is the intersection of essential submodules?

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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Is your ring arbitrary? –  Mariano Suárez-Alvarez Sep 5 '11 at 22:01
I think it is true for non-commutative rings with unit, but I would be happy to prove it for commutative rings. –  ashpool Sep 5 '11 at 22:04