composition series

Let $R$ be a ring and let $M$ be a (right or left) $R$-module.A series of submodules

$$M=M_0\supset M_1\supset M_2\supset \mathrm{\dots }\supset M_n=0$$

in which each quotient $M_i/M_{i+1}$ is simple is called a composition series for $M$.

A module need not have a composition series. For example, the ring of integers, $\mathbb{Z}$, considered as a module over itself, does not have a composition series.

A necessary and sufficient condition for a module to have a composition series is that it is both Noetherian and Artinian.

If a module does have a composition series, then all composition series are the same length.This length (the number $n$ above) is called the composition length of the module.

If $R$ is a semisimple Artinian ring, then $R_R$ and ${}_{R}R$ always have composition series.