---
tag: AG1GMQDTDW
kind: concept
name: Invariant basis number
msc: [16D40, "16E20"]
sources:
  - url: https://en.wikipedia.org/wiki/Invariant_basis_number
same_as:
  wikidata: Q6059504
  wikipedia: Invariant basis number
---

## Summary

A ring has invariant basis number (IBN) if any two bases of a finitely generated [[free module|0DZ9DKAKZR]] over it have the same size.
For such a ring the rank of a free module is well defined, just as the dimension of a vector space is.
All nonzero commutative rings and all division rings have this property, but some noncommutative rings do not.

## Explanations

### L1 human

Over a field, every basis of a vector space has the same number of elements; this number is the dimension.
For modules over a general ring this can fail: a free module can have a basis of $1$ element and also a basis of $2$ elements.
A ring has invariant basis number when this strange behaviour does not happen.
For example, $\Z$ has invariant basis number, so $\Z^m$ and $\Z^n$ are isomorphic only when $m = n$.