Derivations on rank $k$ triangular matrices

Abstract

Let $n\geqslant 2$ be an integer and let $T_n(\mathcal{R})$ be the algebra of $n\times n$ upper triangular matrices over a unital ring $\mathcal{R}$. In this paper, we characterize derivations $\psi:T_n(\mathcal{R})\rightarrow T_n(\mathcal{R})$ on strictly upper triangular matrices, i.e., additive maps $\psi$ satisfying $\psi(AB)=A\psi(B)+\psi(A)B$ for all strictly upper triangular matrices $A,B\in T_n(\mathcal{R})$. We then deduce this result a complete structural characterization of derivations $\psi:T_n(\mathcal{R})\rightarrow T_n(\mathcal{R})$ on rank $k$ upper triangular matrices, where $1\leqslant k\leqslant n$ is a fixed integer and $\mathcal{R}$ is a division ring.