Length of a module

If an $R$-module $M$ admits a composition series

then the length of $M$, denoted $\ell(M)$, is the integer $n$. By the Jordan–Hölder theorem, $\ell(M)$ is well-defined (independent of the chosen composition series) whenever $M$ has at least one composition series; such modules are called “finite length.”

Finite length is tightly linked to chain conditions: modules that are both Noetherian and Artinian have finite length; see Artinian + Noetherian ⇒ finite length .

Examples:

• As a $\mathbb Z$-module, $\ell(\mathbb Z/p^k\mathbb Z)=k$.
• If $V$ is an $n$-dimensional vector space over a field, then $\ell(V)=n$.
• (Nonexample) $\mathbb Z$ has infinite length (no finite composition series).