Length 2004-09-09 21:50:38 2007-08-20 13:15:57 Definition finite length length % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\mc}{\mathcal} \newcommand{\mb}{\mathbb} \newcommand{\mf}{\mathfrak} \newcommand{\ol}{\overline} \newcommand{\ra}{\rightarrow} \newcommand{\la}{\leftarrow} \newcommand{\La}{\Leftarrow} \newcommand{\Ra}{\Rightarrow} \newcommand{\nor}{\vartriangleleft} \newcommand{\Gal}{\text{Gal}} \newcommand{\GL}{\text{GL}} \newcommand{\Z}{\mb{Z}} \newcommand{\R}{\mb{R}} \newcommand{\Q}{\mb{Q}} \newcommand{\C}{\mb{C}} \newcommand{\<}{\langle} \renewcommand{\>}{\rangle} Let $A$ be a ring and let $M$ be an $A$-module. If there is a finite sequence of submodules of $M$ \begin{align*} M=M_0\supset M_1\supset \cdots \supset M_n=0 \end{align*} such that each quotient module $M_i/M_{i+1}$ is simple, then $n$ is necessarily unique by the \PMlinkname{Jordan-H\"older theorem}{JordanHolderDecomposition} for modules. We define the above number $n$ to be the \emph{length} of $M$. If such a finite sequence does not exist, then the length of $M$ is defined to be $\infty$. If $M$ has finite length, then $M$ satisfies both the ascending and descending chain conditions. A ring $A$ is said to have \emph{finite length} if there is an $A$-module whose length is finite.