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.