矩阵代数的乘法映射与反乘法映射

Multiplicative and anti-multiplicative mappings on matrix algebra

设P是一个域,Γn是满足{aEij|i,j=1,2,…,n,a∈P}Γn Mn(P)的一个乘法半群,其中Mn(P)定义P上所有n×n矩阵组成的乘法半群.证明了一个结果:若f∶Γn→Mn(P)是一个保零矩阵的乘法映射,Fij(i,j=1,2,…,n)是Mn(P)中n2个矩阵,且满足FijFkl=δjkFil(i,j,k,l=1,2,…,n),则存在可逆阵S∈Mn(P),使得f(Fij)=S-1FijS,i,j=1,2,…,n.由此刻画了Γn的保迹反乘法映射.

Given field P and multiplicative semigroup Гn satisfying {αEij ｜ i,j = 1,2, ……,n,α∈ p}lahtain in Гn lahtain inMn（P）, where Mn （P） denotes the semigroup of all n×n matrices over P, a result is： suppose that multiplicative mapping f ： Гn→Mn （P） preserves zero matrices, and Fij （i,j = 1,2, ..., n）, the matrices of Mn （P）, satisfy Fij Fld = δjk Fil （ i , j, k, l = 1,2,..., n ） , then there exists an invertible S ∈ Mn（P） such that f（Fij）=S^-1FijS, i,j=1,2,...,n. By the result trace-preserving anti-multiplicative mapping on Гn is characterized.