Zero Traces Of Multiadditive Maps On Invertible Matrices

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).