论文标题
如果$ r^m \ cong r^n $必须$ m = n $？
If $R^m \cong R^n$ must $m=n$?
论文作者
论文摘要
线性代数的基本定理断言，矢量空间的每个基础$ \ mathbb {r}^n $具有$ n $元素。在此说明中，我们提出了W. G. Leavitt的定理，描述了一种不变基数属性在一个人对环上的线性代数而不是在字段上进行线性代数时可能会失败的方式。我们给出了Leavitt定理的证明，将P. M. Cohn和A. L. S.角的思想结合到了一个基本形式，只需要与矩阵和模块化算术相识。
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.
