Consider $M$ an $A$-module. Then given a chain:

we call $n$ the 1;;length of of the chain. We say that a chain of the above type is a composition series if it is maximal. That is, we cannot add a modules anywhere between in the chain. Equivalently, all of the factor modules $M_k/M_{k+1}$ are 1;;simple.

Finally, given an $A$-module $M$, we define the length $\ell (M)$ of $M$ as follows:

1. If a composition series exists, then $\ell (M)$ is the 1;;minimum length of a composition series.
2. If no composition series exists, then define $\ell (M)=\mathrm{\infty }$.