Length of a module

Remarks

Example: for a finite-dimensional vector space $V$ over a field $k$, $\ell_k(V) = \dim_k V$. Indeed, a strict chain of subspaces raises the dimension at each step, and a chain built from a basis has $\dim_k V$ steps. Non-example of finite length: $\ell_\Z(\Z) = \infty$, because $2^n\Z \subsetneq 2^{n-1}\Z \subsetneq \cdots \subsetneq \Z$ has length $n$ for every $n$. A composition series of $M$ is a chain $0 = M_0 \subsetneq \cdots \subsetneq M_n = M$ whose factors $M_i/M_{i-1}$ are simple modules. A module $M$ has finite length if and only if it has a composition series, and then $\ell_R(M) = n$ by the Jordan–Hölder theorem for modules. For a short exact sequence $0 \to M' \to M \to M'' \to 0$, one has $\ell_R(M) = \ell_R(M') + \ell_R(M'')$. The group version of a composition series is the composition series of a group.