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

$$N=U_1\oplus U_2\oplus \mathrm{\dots }\oplus U_n$$

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

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

$\mathrm{u}-\dim M=0$ if and only if $M=0$.