composition series

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


in which each quotient Mi/Mi+1 is simple is called a composition seriesMathworldPlanetmathPlanetmathPlanetmath for M.

A module need not have a composition series. For example, the ring of integers, , 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 NoetherianPlanetmathPlanetmath and ArtinianPlanetmathPlanetmath.

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 compositionMathworldPlanetmathPlanetmath length of the module.

If R is a semisimplePlanetmathPlanetmathPlanetmathPlanetmath Artinian ring, then RR and RR always have composition series.

Title composition series
Canonical name CompositionSeries
Date of creation 2013-03-22 14:04:13
Last modified on 2013-03-22 14:04:13
Owner mclase (549)
Last modified by mclase (549)
Numerical id 6
Author mclase (549)
Entry type Definition
Classification msc 16D10