# socle

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

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

Title socle Socle 2013-03-22 14:21:26 2013-03-22 14:21:26 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 16D99 SemisimpleGroup