Abstract
A fundamental theorem of linear algebra asserts that every basis for the vector space 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.
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