 Length of a module

A module has finite length when it admits a composition series of a module, and its length is the number of simple modules among the successive quotients. The Krull–Schmidt theorem concerns decompositions into indecomposable summands; the independence of composition length instead follows from the Jordan–Hölder theorem. A module without a finite composition series is assigned infinite length.

Length is additive in a short exact sequence: $\ell (M)=\ell (N)+\ell (M/N)$ when the outer modules have finite length. For example, $\mathbb{Z}/p^r\mathbb{Z}$ has length $r$, with each successive factor isomorphic to $\mathbb{Z}/p\mathbb{Z}$. The module $\mathbb{Z}$ has infinite length, as its descending chain $2^n\mathbb{Z}$ never stabilizes.

In the context of module theory, particularly in the realm of algebra, the **length of a module** is a concept used to measure the size and complexity of the module in terms of its composition series. ### Definition: The length of a module \( M \) over a ring \( R \) is defined as the maximum length of a composition series of \( M \).