Strong commutativity preserving additive maps on rank k triangular matrices.

Let $ n\geqslant 2 $ n ⩾ 2 be an integer and let $ T_n({\mathbb {D}}) $ T n (D) be the ring of $ n\times n $ n × n upper triangular matrices over a division ring $ {\mathbb {D}} $ D with centre $ Z(T_n({\mathbb {D}})) $ Z (T n (D)). In this paper, we characterize additive maps $ \psi :T_n({\mathbb {D}})\rightarrow T_n({\mathbb {D}}) $ ψ : T n (D) → T n (D) satisfying $ [\psi (A),\psi (B)]-[A,B]\in Z(T_n({\mathbb {D}})) $ [ ψ (A) , ψ (B) ] − [ A , B ] ∈ Z (T n (D)) for all $ A,B\in T_n({\mathbb {D}}) $ A , B ∈ T n (D). We then deduce from this result a complete characterization of strong commutativity preserving additive maps $ \psi :T_n({\mathbb {D}})\rightarrow T_n({\mathbb {D}}) $ ψ : T n (D) → T n (D) on rank k upper triangular matrices, where $ 1\leqslant k\leqslant n $ 1 ⩽ k ⩽ n is a fixed integer such that $ k\neq n $ k ≠ n when $ |{\mathbb {D}}|=2 $ | D | = 2. [ABSTRACT FROM AUTHOR]