socle

Let $M$ be a left (or right) $R$-module. The socle of $M$, denoted by $\mathrm{Soc}(M)$, is the sum of all simple (or minimal) submodules of $M$. If $M$ has no simple submodules, then $\mathrm{Soc}(M):=(0)$.

A left (or right) $R$-module $M$ is semisimple iff $\mathrm{Soc}(M)=M$.

Title	socle
Canonical name	Socle
Date of creation	2013-03-22 14:21:26
Last modified on	2013-03-22 14:21:26
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	5
Author	CWoo (3771)
Entry type	Definition
Classification	msc 16D99
Related topic	SemisimpleGroup