# 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 |