保持交换环上矩阵若干性质的映射

Mappings That Preserve Properties of Matrices over Commutative Rings

将域上的矩阵保持问题推广到交换环,有条件地得到了类似于域上的矩阵保持问题的若干结论,刻画了交换环上的矩阵空间中保持三类性质的映射:1)上三角矩阵空间的保持幂等的映射;2)全矩阵空间和上三角矩阵空间的保持对合的映射;3)全矩阵空间、上三角矩阵空间和对称矩阵空间的保持正交性的映射.

We generalize preserving problems of matrices over fields to those over commutative rings and extend several existing results for matrices over fields. We characterize three classes of mappings that preserve certain properties of matrices over a commutative ring： 1） mappings preserving idempotence in upper triangular matrix spaces; 2） mappings preserving involution in full matrix spaces and upper triangular matrix spaces; 3） mappings preserving orthogonality in full matrix spaces, upper triangular matrix spaces, and symmetric matrix spaces.