Socle of a module

Remarks

Example: let $M = \Z/12\Z$ as a $\Z$-module. A submodule is simple exactly when it has prime order, so the simple submodules are $6\Z/12\Z$ and $4\Z/12\Z$, of orders $2$ and $3$. Their sum is $2\Z/12\Z$, so $\operatorname{soc}(\Z/12\Z) = 2\Z/12\Z$.

Non-example of a nonzero socle: the $\Z$-module $\Z$ has socle $0$. Each nonzero submodule $n\Z$ contains the proper nonzero submodule $2n\Z$, so no submodule of $\Z$ is simple.

The socle is a semisimple module, and it contains every semisimple submodule of $M$. It also equals the intersection of all essential submodules of $M$. The socle of the left module ${}_R R$ is a two-sided ideal of $R$.