如果Rm = Rn必须m = n?

Pub Date : 2023-03-21 DOI:10.1080/00029890.2023.2184163

Tyrone Crisp

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引用次数: 0

摘要

线性代数的一个基本定理断言向量空间Rn的每一组基有n个元素。在这篇说述性的笔记中，我们提出了W. G. Leavitt的一个定理，该定理描述了当人们在环上而不是在域上做线性代数时，这个不变基数性质可能失效的一种方式。我们给出了莱维特定理的一个证明，这个证明结合了P. M. Cohn和a . L. S. Corner的思想，形成了一个初等的形式，只需要对矩阵和模运算略知一知。

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If R^m ≅ Rⁿ must m = n?

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.

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