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\(m\)-power commuting maps on semiprime rings. (English) Zbl 1296.16047

Let \(R\) be a semiprime ring with center \(Z(R)\), extended centroid \(C\), and maximal right quotient ring \(U\). An additive \(f\colon R\to U\) is called commuting if for all \(r\in R\), \([f(r),r]=f(r)r-rf(r)=0\), and is called centralizing when all \([f(r),r]\in C\). For an integer \(m>0\), \(f\) is \(m\)-power commuting (or centralizing) if \([f(r),r^m]=0\) (or \([f(r),r^m]\in C\)).
The main result of the authors is that if a \(Z(R)\)-linear \(f\colon R\to U\) is \(m\)-power commuting then there is \(e^2=e\in C\) so that on \(R\), \(ef(r)=cr+\mu(r)\), where \(c\in C\) and \(\mu\colon R\to C\), and \((1-e)U\cong M_2(E)\) for \(E\) a complete Boolean ring. Two consequences of this result are that if \(f\) is 2-power commuting then \(f\) is commuting, and if \(f\) is centralizing, then \(f\) is commuting.

MSC:

16W20 Automorphisms and endomorphisms
16R60 Functional identities (associative rings and algebras)
16N60 Prime and semiprime associative rings
16W25 Derivations, actions of Lie algebras
16D50 Injective modules, self-injective associative rings
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