# 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 $M$

 $\displaystyle M=M_{0}\supset M_{1}\supset\cdots\supset M_{n}=0$

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 $\infty$.

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 LengthOfAModule 2013-03-22 14:35:32 2013-03-22 14:35:32 CWoo (3771) CWoo (3771) 11 CWoo (3771) Definition msc 16D10 msc 13C15 finite-length module finite length