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

Let F be a field, let N∞( F) be the ring of all N× N strictly upper triangular matrices over F, and let T∞( F) be the ring of all N× N upper triangular matrices over F. In this paper, we completely characterize additive maps f:N∞( F)→ T∞( F) satisfying $[f(x),x]=0$ for all x∈ N∞( 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].