Maps determined by rank- $\boldsymbol{s}$ s matrices for relatively small $\boldsymbol{s}$ s

Abstract

Let n and s be integers such that 1≤s<n21\le s<\frac{n}{2}, and let Mn(K)M_n(\mathbb {K}) be the ring of all n×nn\times n matrices over a field K\mathbb {K}. Denote by [ns][\frac{n}{s}] the least integer m with m≥nsm\ge \frac{n}{s}. In this short note, it is proved that if g:Mn(K)→Mn(K)g:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K}) is a map such that g(∑i=1[ns]Ai)=∑i=1[ns]g(Ai)g\left( \sum _{i=1}^{[\frac{n}{s}]}A_i\right) =\sum _{i=1}^{[\frac{n}{s}]}g(A_i) holds for any [ns][\frac{n}{s}] rank-s matrices A1,…,A[ns]∈Mn(K)A_1,\ldots ,A_{[\frac{n}{s}]}\in M_n(\mathbb {K}), then g(x)=f(x)+g(0), x∈Mn(K)x\in M_n(\mathbb {K}), for some additive map f:Mn(K)→Mn(K)f:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K}). Particularly, g is additive if charK∤([ns]−1)char\mathbb {K}\not \mid \left( [\frac{n}{s}]-1\right) .