Length of a module - Knowlpedia

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

0=M_0 \subset M_1 \subset \cdots \subset M_n=M,

then the length of $M$ , denoted $\ell_R(M)$ , is $n$ . The Jordan–Hölder theorem implies that this number is independent of the chosen composition series. A module admitting such a series is a finite-length module.

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, $\ell_{\mathbb Z}(\mathbb Z/p^k\mathbb Z)=k$ .
If $V$ is an $n$ -dimensional vector space over a field $K$ , then $\ell_K(V)=n$ .
The $\mathbb Z$ -module $\mathbb Z$ has no finite composition series.