﻿ Additive maps determined by rank-$s$ matrices-东北师范大学数学与统计学院
Additive maps determined by rank-$s$ matrices

Let $M_n(\mathbb{K})$ be the ring of all $n\times n$ matrices over a field $\mathbb{K}$, where $n\geq 2$ is a fixed integer. We proved that a map $f:M_n(\mathbb{K})\rightarrow M_n(\mathbb{K})$ is  additive if and only if $f(A+B)=f(A)+f(B)$ for all rank-$s$ matrices $A,B\in M_n(\mathbb{K})$,where $s$ is a fixed positive integer such that $\frac{n}{2}\leq s\leq n$. For the case $1\leq s<\frac{n}{2}$, we also 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})$, where$[\frac{n}{s}]$denotes the least integer$m$with$m\geq \frac{n}{s}$. Particularly,$g$is additive if$char\mathbb{K}\nmid \left([\frac{n}{s}]-1\right)\$.