Let \(\mathcal R\) be a ring. The mapping \(f\colon{\mathcal R}\to{\mathcal R}\) is commuting if \([f(x),x]=0\) for all \(x\in{\mathcal R}\). The study of commuting and centralizing mappings (when \([f(x),x]\) is in the centre of \(\mathcal R\)) was initiated by E. C. Posner [Proc. Am. Math. Soc. 8, 1093-1100 (1958; Zbl 0082.03003)] who related derivations and commutativity conditions.
The main result of the paper under review is the following. Let \({\mathcal T}_r\), \(r\geq 2\), be the algebra of \(r\times r\) upper triangular matrices over a field \(\mathcal F\) and let \(f\colon{\mathcal F}_r^n\to{\mathcal F}_r\) be a multilinear mapping such that \([f(A,\dots,A),A]=0\) for all \(A\in{\mathcal T}_r\). If \(n\leq r\) and \(|{\mathcal F}|>n+1\), then there exist multilinear mappings \(\lambda_i\colon{\mathcal F}_r^i\to{\mathcal F}_r\) such that \[ f(A,\dots,A)=\sum_{i=0}^n\lambda_i(A,\dots,A)A^{n-i} \] for all \(A\in{\mathcal T}_r\). As a consequence the authors describe the bijective linear mappings \(\theta\colon{\mathcal F}_r\to{\mathcal F}_r\), \(r\geq 3\), satisfying \([\theta(A^2),\theta(A)]=0\) for all \(A\in{\mathcal T}_r\) (assuming that \(\text{char}({\mathcal F})\not=2\) and \(|{\mathcal F}|>3\)). It is worth to mention that the paper contains a long list of references.