composition series

Let $R$ be a ring and let $M$ be a (right or left) $R$ -module. A series of submodules $$M = M_0 \supset M_1 \supset M_2 \supset \dots \supset M_n = 0$$ 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 ${}_RR$ always have composition series.