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Higher derivations of semiprime rings. (English) Zbl 1010.16028
A family of additive maps $$(d_i)_{i\geq 0}$$ of a ring $$R$$ is called a higher derivation if $$d_0=\text{id}_R$$ and $$d_n(ab)=\sum^n_{i=0}d_i(a)d_{n-i}(b)$$ for all $$a,b\in R$$ and $$n\geq 0$$. For example, if $$R$$ is an algebra over $$\mathbb{Q}$$ and $$d$$ is a derivation of $$R$$, then $$(\tfrac{d^i}{i!})_{i\geq 0}$$ is a higher derivation. The main topic of the paper is the condition $$\sum^n_{i=0}a_id_i(x)=0$$ for all $$x\in R$$, where the $$a_i$$’s are some fixed elements, $$(d_i)$$ is a higher derivation, and $$R$$ is a semiprime ring. Roughly speaking, under certain assumptions the authors describe the structure of a higher derivation satisfying this condition. In particular, some results can be considered as generalizations of V. K. Kharchenko’s well-known theorem on algebraic derivations of prime rings [Algebra Logika 17, 220-238 (1978; Zbl 0423.16011)].

##### MSC:
 16W25 Derivations, actions of Lie algebras 16N60 Prime and semiprime associative rings 16R50 Other kinds of identities (generalized polynomial, rational, involution)
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