uniform dimension

Let M be a module over a ring R, and suppose that M contains no infinite direct sumsPlanetmathPlanetmathPlanetmath of non-zero submodules. (This is the same as saying that M is a module of finite rank.)

Then there exists an integer n such that M contains an essential submodule N where


is a direct sum of n uniform submodules.

This number n does not depend on the choice of N or the decomposition into uniform submodules.

We call n the uniform dimension of M. Sometimes this is written u-dimM=n.

If R is a field K, and M is a finite-dimensional vector spaceMathworldPlanetmath over K, then u-dimM=dimKM.

u-dimM=0 if and only if M=0.

Title uniform dimension
Canonical name UniformDimension
Date of creation 2013-03-22 14:02:59
Last modified on 2013-03-22 14:02:59
Owner mclase (549)
Last modified by mclase (549)
Numerical id 7
Author mclase (549)
Entry type Definition
Classification msc 16P60
Related topic GoldieRing