length of a module

Let A be a ring and let M be an A-module. If there is a finite sequence of submodulesMathworldPlanetmath of M


such that each quotient moduleMathworldPlanetmath Mi/Mi+1 is simple, then n is necessarily unique by the Jordan-Hölder theorem (http://planetmath.org/JordanHolderDecomposition) for modules. We define the above number n to be the length of M. If such a finite sequence does not exist, then the length of M is defined to be .

If M has finite length, then M satisfies both the ascending and descending chain conditionsMathworldPlanetmathPlanetmathPlanetmath.

A ring A is said to have finite length if there is an A-module whose length is finite.

Title length of a module
Canonical name LengthOfAModule
Date of creation 2013-03-22 14:35:32
Last modified on 2013-03-22 14:35:32
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 16D10
Classification msc 13C15
Synonym finite-length module
Defines finite length