【作者】 朱建伟；

【导师】 徐晓伟；

【作者基本信息】 吉林大学 ， 基础数学， 2019， 硕士

【摘要】 设Mn(K)为域K上n× n矩阵环,Mns(K)为域K上秩为s的n × n矩阵集,GLn(K)为K上n × n可逆矩阵集.记[n/s]为使m ≥n/s的最小整数m.对于一个结合环R,一个映射G:(?)被称作m重可加的,如果对任意i∈ {1,2,..,m}及任意的 x1,…,xi-1,xi,xi,xi+1,…,xm∈R,都有G(x1,… xi-1,xi+xi,xi+1,…,xm)=G(x1,…,xi-1,xi,xi+1,…xm)+G(x1,…xi-1,xi,xi+1,…,xm).由T(x)=G(x,x,…,x)定义的映射T:R→R被称作G的迹.本文主要讨论了Mns(K)上加性映射的性质,以及m重加性映射迹函数为零映射的一个充要条件.本文证明了:(1)设n,s为整数,且满足1 ≤s<n/2,如果g:Mn(K)→ Mn(K)是一个映射使得对任意[n/s]个秩为s的矩阵A1,…,A[n/s]∈Mn(K),g(?)=(?)(Ai)成立,则存在一个加性映射/:Mn(K)→ Mn(K),使得g(x)=f(x)+g(0),x∈ Mn(K).特别地,如果charK([n/s]-1),则g可加.(2)设n,m均为大于1的整数,假设G:Mn(K)m→Mn(K)是m重可加映射且满足对任意x ∈ GLn(K),G(x,x,…,x)=0.那么对任意x ∈ Mn(X)如果下列一项成立:(1)charK=0;(2)char K>m;(3)char K=m 且 |K|≠n;(4)|K|≥ 2m,则G(x,x,…,x)=0.作为应用,得到了Franca定理,且Beidar等人的结论对素环Mn(K)的子集GLn(K)仍然成立.更多还原

【Abstract】 Let Mn(K)be the ring of all n×n matrices over a field K,Msn(K)be the ring of all n×n rank-s matrices over a field K.Denote by[n/s]the least integer m with m>For an associative ring R,a map G:(?)R is called m-additive if for each fixed G(x1,…xi-1,xi+xi,xi+1,…,xm)=G(x1,…,xi-1,xi,xi+1,…,xm)+G(x1,…,xi-1,xi,xi+1,…,xm)holds for all x1,…,xi-1,xi,zi,zi+1,…,xm∈R.The map T:R→R defined by T(x)=G(x,x,…,x)is called the trace of G.This paper mainly discusses the properties of the additive maps on the Mns(K),and the sufficient condition for the trace of an m-additive map to be zero map.In this paper,it is proved that(1)Let n and s be integers such that 1 ≤s<n/2,if g:Mn(K)→ Mn(K)is a map such that g(?)=(?)holds for any[n/s]rank-s matrices A1,…,A[n/s]∈Mn(K),then g(x)=f(x)+g(0),x∈ Mn(K),for some additive map f:Mn(K)Mn(K).Particularly,g is additive if charK |([n/s]-1).(2)Let n,m>1 be integers,and let M {K)be the ring of all n ×n matrices over a field K.Assume that G:Mn(K)m→ Mn(K)is an m-additive map such that G(x,…,x)=0 for all x ∈ GLn(K),the set of all n x n invertible matrices over K.Then G(x,x,…,x)=0 for all x E M(K)if one of the followings holds:(1)charK=0;(2)charK>m;(3)charK=m and|K|≠m;(4)|K|≥2m.As applications,the theorem of Franca is obtained and the theorem by Beidar et al.holds for the subset GLn(K)of the prime ring Mn(K).更多还原