# Multiplicative derivations on rank-$s$ matrices for relatively small $s$

## TL;DR

This paper characterizes maps on matrix rings that behave like derivations on rank-$s$ matrices, showing they are essentially derivations on all matrices up to rank $s$, for fixed $n$ and $s$.

## Contribution

It establishes that maps satisfying a derivation-like property on rank-$s$ matrices are actually derivations on all matrices of rank up to $s$, extending the understanding of multiplicative derivations.

## Key findings

- Maps satisfying the derivation property on rank-$s$ matrices are derivations on those matrices.
- Such maps coincide with derivations on all matrices of rank up to $s$.
- The result holds for fixed $n$ and $s$ with $1 \\leq s \\leq \\frac{n}{2}$.

## Abstract

Let $n$ and $s$ be fixed integers such that $n\geq 2$ and $1\leq s\leq \frac{n}{2}$. Let $M_n(\mathbb{K})$ be the ring of all $n\times n$ matrices over a field $\mathbb{K}$. If a map $\delta:M_n(\mathbb{K})\rightarrow M_n(\mathbb{K})$ satisfies that $\delta(xy)=\delta(x)y+x\delta(y)$ for any two rank-$s$ matrices $x,y\in M_n(\mathbb{K})$, then there exists a derivation $D$ of $M_n(\mathbb{K})$ such that $\delta(x)=D(x)$ holds for each rank-$k$ matrix $x\in M_n(\mathbb{K})$ with $0\leq k\leq s$.