Commuting additive maps on upper triangular and strictly upper triangular infinite matrices

Abstract

Let ${\mathbb F}$ be a field, let $N_{\infty}({\mathbb F})$ be the ring of all ${\mathbb N}\times {\mathbb N}$ strictly upper triangular matrices over ${\mathbb F,}$ and let $T_{\infty}({\mathbb F})$ be the ring of all ${\mathbb N}\times {\mathbb N}$ upper triangular matrices over ${\mathbb F}$. In this paper, we completely characterize additive maps $f:N_{\infty}({\mathbb F})\to T_{\infty}({\mathbb F})$ satisfying $[f(x),x]=0$ for all $x\in N_{\infty}({\mathbb F})$. As applications, we obtain the finite fields versions of the two main results recently obtained by Slowik and Ahmed [Electron. J. Linear Algebra 37:247-255, 2021].