2022/03/24 by Tyrone Crisp, Crisp, Tyrone
Mathematics · #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA
paper · pdf · doi:10.48550/arxiv.2203.13155
7 pages, expository note
arxiv created 2022/03/24 · arxiv updated 2022/03/25
A fundamental theorem of linear algebra asserts that every basis for the vector space ℝn 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.