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

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

$\mathrm{u}-\mathrm{dim}M=0$ 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 |