---
tag: S2KXCAH5MM
kind: concept
name: Socle of a module
msc: [16D60, 16D70]
sources:
  - url: https://en.wikipedia.org/wiki/Socle_(mathematics)
---

## Summary

The socle of a module is the sum of all its [[simple submodules|G00C22AEDH]].
It is the largest [[semisimple submodule|N3MRHDKDGC]] of the module.
It measures the part of a module that is built directly from simple pieces.
It is a basic tool in the study of Artinian rings and injective modules.

## Explanations

### L1 human

Simple modules are the atoms of module theory: they have no proper nonzero submodules.
The socle collects every atom that sits at the bottom of a module.
Picture a module as a building: the socle is its ground floor.
For example, in the $\Z$-module $\Z/12\Z$ the simple submodules are $\set{0, 6}$ and $\set{0, 4, 8}$, and their sum is the socle, the subgroup generated by $2$, which has $6$ elements.