# composition series

Let $R$ be a ring and let $M$ be a (right or left) $R$-module. A series of submodules

$$M={M}_{0}\supset {M}_{1}\supset {M}_{2}\supset \mathrm{\dots}\supset {M}_{n}=0$$ |

in which each quotient ${M}_{i}/{M}_{i+1}$ is simple is called a composition series^{} for $M$.

A module need not have a composition series. For example, the ring of integers, $\mathbb{Z}$, considered as a module over itself, does not have a composition series.

A necessary and sufficient condition for a module to have a composition series is that it is both Noetherian^{} and Artinian^{}.

If a module does have a composition series, then all composition series are the same length.
This length (the number $n$ above) is called the *composition ^{} length* of the module.

If $R$ is a semisimple^{} Artinian ring, then ${R}_{R}$ and ${}_{R}R$ always have composition series.

Title | composition series |
---|---|

Canonical name | CompositionSeries |

Date of creation | 2013-03-22 14:04:13 |

Last modified on | 2013-03-22 14:04:13 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 6 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 16D10 |