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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)].

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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