EAS除环上矩形矩阵的华氏几何基本定理
Hua's fundamental theorem of geometry of rectangular matrices over EAS division rings
摘要：
矩形矩阵几何的基本定理描述了除环$\mathbb{D}$上所有$m\times n$矩阵空间上保持两个方向邻接的双射映射的一般形式。华在二十世纪四十年代证明的这个结果最近在几个方向上得到了改进。人们可以在没有双射性假设的情况下研究这样的映射，或者可以在仅在一个方向上保持邻接的较弱假设下尝试得到相同的结论。最后一种可能性是研究作用在不同大小的矩阵空间之间的映射。最佳结果将描述在没有任何正则性条件（内射性或满射性）的情况下，仅在不同大小的矩形矩阵空间之间作用的保持单向邻接的映射。一个除环称为EAS，如果它不同构于任何真子环.以前已经知道，可以在非EAS的除环上的矩阵上构造具有野生行为的邻接保持映射。对于EAS除环上的矩阵，最近证明了作用在满足某种弱满射条件的不同大小的矩阵空间之间的保邻映射是退化的或具有期望的简单标准式.我们将删除这个弱满射性假设，从而完全解决长期悬而未决的问题的最佳版本的华定理。
英文摘要：
The fundamental theorem of geometry of rectangular matrices describes the general form of bijective maps on the space of all $m\times n$ matrices over a division ring $\mathbb{D}$ which preserve adjacency in both directions. This result proved by Hua in the nineteen forties has been recently improved in several directions. One can study such maps without the bijectivity assumption or one can try to get the same conclusion under the weaker assumption that adjacency is preserved in one direction only. And the last possibility is to study maps acting between matrix spaces of different sizes. The optimal result would describe maps preserving adjacency in one direction only acting between spaces of rectangular matrices of different sizes in the absence of any regularity condition (injectivity or surjectivity). A division ring is said to be EAS if it is not isomorphic to any proper subring. It has been known before that it is possible to construct adjacency preserving maps with wild behaviour on matrices over division rings that are not EAS. For matrices over EAS division rings it has been recently proved that adjacency preserving maps acting between matrix spaces of different sizes satisfying a certain weak surjectivity condition are either degenerate or of the expected simple standard form. We will remove this weak surjectivity assumption, thus solving completely the long standing open problem of the optimal version of Hua's theorem.
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