If $R^m \cong R^n$ must $m=n$?

TL;DR

This paper explores conditions under which the invariant basis number property fails in linear algebra over rings, contrasting with the classical case over fields where dimension is uniquely determined.

Contribution

It presents a proof of Leavitt's theorem demonstrating how the invariant basis number property can fail over rings, using elementary matrix and modular arithmetic concepts.

Findings

The invariant basis number property does not always hold over rings.

Leavitt's theorem provides conditions for failure of basis invariance.

Elementary proof combines ideas from Cohn and Corner.

Abstract

A fundamental theorem of linear algebra asserts that every basis for the vector space $\mathbb{R}^n$ has $n$ elements. In this expository note we present a theorem of W. G. Leavitt describing one way in which this invariant basis number property can fail when one does linear algebra over rings, rather than over fields. We give a proof of Leavitt's theorem that combines ideas of P. M. Cohn and A. L. S. Corner into an elementary form requiring only a nodding acquaintance with matrices and modular arithmetic.