1
$\begingroup$

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?

  • 0
    I think it is true for non-commutative rings with unit, but I would be happy to prove it for commutative rings.2011-09-05

1 Answers 1

2

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

  • 0
    Dead link. ${}$2018-08-15