On centralizing additive maps on rank one triangular matrices over division rings

Abstract

Let $n\geqslant 2$ be an integer and let ${\mathbb {D}}$ be a division ring with center $Z({\mathbb {D}}).$ We denote by $T_n({\mathbb {D}})$ the ring of $n\times n$ upper triangular matrices over ${\mathbb {D}}$ with unity $I_n.$ We characterize commuting additive maps $\psi :T_n({\mathbb {D}})\rightarrow T_n({\mathbb {D}})$ on rank one matrices over a noncommutative division ring ${\mathbb {D}}$ and show that there exists $\lambda \in Z({\mathbb {D}}),$ an additive map $\mu :T_n({\mathbb {D}})\rightarrow Z({\mathbb {D}})$, and a set of additive maps ${\mathscr {F}}=\bigcup _{1<s\leqslant t<n}\{\phi ^{(s,t)}_{ij}:{\mathbb {D}}\rightarrow {\mathbb {D}}\,|\,1\leqslant i\leqslant s-1 \,\ \textrm{and} \,\ t+1\leqslant j\leqslant n\}$, such that

$$\begin{aligned} \psi (A)=\lambda A+\mu (A)I_n+\psi _{{\mathscr {F}}}(A) \end{aligned}$$

for all $A\in T_n({\mathbb {D}}),$ where $\psi _{\mathscr {F}}:T_n({\mathbb {D}})\rightarrow T_n({\mathbb {D}})$ is the additive map defined by

$$\begin{aligned} \psi _{{\mathscr {F}}}(A)=\sum _{1<s\leqslant t<n}\left( \sum _{i=1}^{s-1}\sum _{j=t+1}^n\phi ^{(s,t)}_{ij}(a_{st})E_{ij}\right) \end{aligned}$$

for all $A=(a_{ij})\in T_n({\mathbb {D}})$ and $\psi _{{\mathscr {F}}}=0$ when $n=2.$ Here, $E_{ij}\in T_n({\mathbb {D}})$ is the matrix whose (i, j)th entry is one and zero elsewhere. We then deduce from this result a complete description of centralizing additive maps $\psi :T_n({\mathbb {D}})\rightarrow T_n({\mathbb {D}})$ on rank one matrices over division rings ${\mathbb {D}}.$