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?
Socle is the intersection of essential submodules?
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$\begingroup$
abstract-algebra
commutative-algebra
modules
socle
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1Is your ring arbitrary? – 2011-09-05
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0I 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
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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
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1A hint version might be to prove a module is semi-simple iff it has no proper essential submodules. (Again the direction you've already done is the easy one since direct summands are not usually essential, and the other direction looks at maximal complements as in the notes). – 2011-09-05
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1Thanks! I also found a nice proof in Rings and Categories of modules, Frank Anderson & Kent Fuller. – 2011-09-05
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0Dead link. ${}$ – 2018-08-15