Composition series

A composition series of an $R$-module $M$ is a finite chain of submodules

such that each factor $M_i/M_{i-1}$ is a simple module (a simple quotient in the sense of quotient modules ). The integer $n$ is the length of the series and, when a composition series exists, is an invariant of $M$ (Jordan–Hölder), recorded as the module length .

Composition series exist exactly for modules of finite length and provide a canonical way to “factor” modules into simple pieces.

Examples:

• For the $\mathbb Z$-module $\mathbb Z/p^k\mathbb Z$, the chain $0\subset p^{k-1}\mathbb Z/p^k\mathbb Z \subset \cdots \subset p\mathbb Z/p^k\mathbb Z \subset \mathbb Z/p^k\mathbb Z$ is a composition series.
• For an $n$-dimensional vector space $V$ over a field, any flag $0\subset V_1\subset\cdots\subset V_n=V$ with $\dim(V_i)=i$ is a composition series.
• (Nonexample) The $\mathbb Z$-module $\mathbb Z$ has no composition series.