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

Abstract

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