# length of a module

Let $A$ be a ring and let $M$ be an $A$-module. If there is a finite sequence of submodules^{} of $M$

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

such that each quotient module^{} ${M}_{i}/{M}_{i+1}$ is simple, then $n$ is necessarily unique by the Jordan-Hölder theorem (http://planetmath.org/JordanHolderDecomposition) for modules. We define the above number $n$ to be the *length* of $M$. If such a finite sequence does not exist, then the length of $M$ is defined to be $\mathrm{\infty}$.

If $M$ has finite length, then $M$ satisfies both the ascending and descending chain conditions^{}.

A ring $A$ is said to have *finite length* if there is an $A$-module whose length is finite.

Title | length of a module |
---|---|

Canonical name | LengthOfAModule |

Date of creation | 2013-03-22 14:35:32 |

Last modified on | 2013-03-22 14:35:32 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 11 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 16D10 |

Classification | msc 13C15 |

Synonym | finite-length module |

Defines | finite length |