Skew-commuting and commuting mappings in rings
Let \(R\) be a ring with center \(Z\), and \(S\) a nonempty subset of \(R\); and let \(n\) be a fixed positive integer. A mapping \(f\colon R\to R\) is called \(n\)-commuting (resp. \(n\)-centralizing) on \(S\) if \([x^n,f(x)]=0\) (resp. \([x^n,f(x)]\in Z\)) for all \(x\in S\). Similarly, \(f\) is \(n\)-skew-commuting (resp. \(n\)-skew-centralizing) on \(S\) if \(x^nf(x)+f(x)x^n=0\) (resp. \(x^nf(x)+f(x)x^n\in Z\)) for all \(x\in S\). Motivated by results of the reviewer and \textit{J. Lucier} [Result. Math. 36, No. 1-2, 1-8 (1999; Zbl 0938.16027)], the authors prove several theorems on rings admitting maps \(f\) with one of the above commuting or centralizing conditions on an additive subgroup \(H\) containing a left identity element \(e\). For example, they deduce the following result: Let \(n\geq 2\), and let \(R\) be an \(n!\)-torsion-free ring with left identity \(e\), and let \(H\) be an additive subgroup of \(R\) containing \(e\). Let \(G\colon R\times R\to R\) be a symmetric biadditive map with trace \(g\). If \(g\) is \(n\)-skew-commuting on \(H\), then \(g(H)=\{0\}\). In the final section, the authors generalize a result of \textit{J. Vukman} [Bull. Aust. Math. Soc. 53, No. 3, 353-359 (1996; Zbl 0883.16026)] as follows: Let \(n\geq 2\), and let \(R\) be an \(n!\)-torsion-free semiprime ring. If \(d\) is a derivation on \(R\) such that \(d^2\) is \(n\)-commuting on \(R\), then \(d(R)\subseteq Z\).
scientific article; zbMATH DE number 1837086