If Rm ≅ Rn must m = n?
IF 0.4
4区 数学
Q4 MATHEMATICS
引用次数: 0
Abstract
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.如果Rm = Rn必须m = n?
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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来源期刊
American Mathematical Monthly
Mathematics-General Mathematics
CiteScore
0.80
自引率
20.00%
发文量
127
审稿时长
6-12 weeks
期刊介绍：
The Monthly''s readers expect a high standard of exposition; they look for articles that inform, stimulate, challenge, enlighten, and even entertain. Monthly articles are meant to be read, enjoyed, and discussed, rather than just archived. Articles may be expositions of old or new results, historical or biographical essays, speculations or definitive treatments, broad developments, or explorations of a single application. Novelty and generality are far less important than clarity of exposition and broad appeal. Appropriate figures, diagrams, and photographs are encouraged.
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