# 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 UniformDimension 2013-03-22 14:02:59 2013-03-22 14:02:59 mclase (549) mclase (549) 7 mclase (549) Definition msc 16P60 GoldieRing