uniform dimension

Let $M$ be a module over a ring $R$ , and suppose that $M$ contains no infinite direct sums 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

N=U_{1}\oplus U_{2}\oplus\dots\oplus U_{n}

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 $\operatorname{u-dim}M=n$ .

If $R$ is a field $K$ , and $M$ is a finite-dimensional vector space over $K$ , then $\operatorname{u-dim}M=\dim_{K}M$ .

$\operatorname{u-dim}M=0$ 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