交换整环上保持矩阵相似或合同关系的函数  

Functions preserving similarity and congruence relations in matrix spaces over a commutative domain

作　　者：刘一宣 谭宜家[1] LIU Yi-xuan;TAN Yi-jia(College of Mathematics and Statistics,Fuzhou University,Fuzhou 350108,China)

机构地区：[1]福州大学数学与统计学院,福州350108

出　　处：《哈尔滨商业大学学报（自然科学版）》2022年第6期763-768,共6页Journal of Harbin University of Commerce：Natural Sciences Edition

基　　金：国家自然科学基金面上项目(No.11971111);福建省自然科学基金面上项目(No.2016J01012)。

摘　　要：探讨了交换整环上矩阵空间保持相似关系和对称矩阵空间保持合同关系的函数,证明了如下结论:设f是交换整环R到自身的一个映射,且当f(1)≠0时f(1)可逆,n(n≥3)是一个整数.则有:1)f是R上n阶矩阵空间的保持相似关系的函数当且仅当f≡c,其中c∈R或f=cδ,其中c∈R且c≠0,δ为R的非零自同态;2)f是R上n阶对称矩阵空间的保持合同关系的函数当且仅当f≡c,其中c∈R或f=cδ,其中c∈R,c≠0,δ为R的非零自同态.The functions of preserving similarity in the matrix spaces and preserving congruence relations in the symmetric matrix spaces over a commutative domain were characterized in this paper,and the following result was proved:let f be a mapping of commutative domain R to itself satisfying f(1)was invertible in R for f(1)≠0 and n,(n≥3),an integer.Then f was a function of preserving similarity in the n-order matrix space over R if and only if f≡c for some c∈R or f=cδfor some c∈R with c≠0 andδwas a nonzero endomorphism of R,and f was a function of preserving congruence relations in the n-order symmetric matrix space over R if and only if f≡c for some c∈R or f=cδfor some c∈R with c≠0 andδwas a nonzero endomorphism of R.

关 键 词：矩阵空间 对称矩阵空间 相似关系 合同关系 保持问题 交换整环 

分 类 号：O153.3[理学—数学]