---
tag: YXEVE39Q31
kind: concept
name: Uniform module
msc: [16D10, 16P60]
sources:
  - url: https://en.wikipedia.org/wiki/Uniform_module
same_as:
  wikidata: Q7885110
  wikipedia: Uniform module
---

## Summary

A uniform module is a nonzero [[module|0NZ8JA25WK]] in which any two nonzero submodules intersect in a nonzero submodule.
Equivalently, every nonzero submodule is an [[essential submodule|8MS51FASMX]].
Uniform modules are the units for measuring the uniform (Goldie) dimension of a module, which counts how many independent uniform submodules fit inside it.
They are a main tool in the proof of Goldie's theorem on rings of quotients of semiprime Noetherian rings.

## Explanations

### L1 human

In a uniform module, nonzero pieces can never be disjoint: any two of them share a nonzero element.
So the module cannot contain two independent directions, as $\Z \oplus \Z$ does.
The integers form a uniform $\Z$-module, since for nonzero $a$ and $b$ the subgroups $a\Z$ and $b\Z$ share $ab \ne 0$.
Every uniform module is [[indecomposable|E2P29GCD9T]], since two summands of a direct sum meet only in zero.