## 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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