Uniserial module

Remarks

Example: for a prime $p$ and $n \ge 1$, the $\Z$-module $\Z/p^n\Z$ is uniserial. Its submodules are the subgroups $p^i\Z/p^n\Z$ for $0 \le i \le n$, and $p^i\Z/p^n\Z \supseteq p^j\Z/p^n\Z$ when $i \le j$. In the same way, $k[x]/(x^n)$ is uniserial as a module over itself, for a field $k$.

Non-example: the $\Z$-module $\Z$ is not uniserial. Its submodules $2\Z$ and $3\Z$ are not comparable, since $2 \notin 3\Z$ and $3 \notin 2\Z$.

A nonzero uniserial module is indecomposable, since two nonzero submodules always meet in the smaller one, which is nonzero. A uniserial module of finite length has exactly one composition series.