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On some equations related to derivations in rings. (English) Zbl 1100.16031

If \(R\) is a ring and \(D\) a derivation on \(R\) such that \(D(x)x=xD(x)\) for all \(x\in R\), then \(D(x^{n+m+1})=(n+m+1)x^mD(x)x^n\) for all \(x\in R\) and all nonnegative integers \(m,n\). Motivated by this observation, the authors establish the following: Let \(m,n\geq 0\) with \(m+n\neq 0\), and let \(R\) be an \((m+n+2)!\)-torsionfree semiprime ring with \(1\). If \(D\) is an additive map on \(R\) such that \(D(x^{m+n+1})=(m+n+1)x^mD(x)x^n\) for all \(x\in R\), then \(D\) is a derivation and \(D(R)\) is central. Moreover, if \(R\) is prime and not commutative, then \(D=0\).
The authors also state an interesting conjecture: If \(R\) is a semiprime ring with suitable torsion restrictions, and \(D\) is an additive map such that \((m+n)D(x^2)=2nD(x)x+2mxD(x)\) for all \(x\in R\) and some \(m,n\geq 0\) with \(m+n\neq 0\neq m-n\), then \(D\) is a derivation mapping \(R\) into its center.

MSC:

16W25 Derivations, actions of Lie algebras
16N60 Prime and semiprime associative rings
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