# 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 ${}_{R}R$ always have composition series.

Title composition series CompositionSeries 2013-03-22 14:04:13 2013-03-22 14:04:13 mclase (549) mclase (549) 6 mclase (549) Definition msc 16D10