an:01796051
Zbl 1010.16028
Ferrero, Miguel; Haetinger, Claus
Higher derivations of semiprime rings
EN
Commun. Algebra 30, No. 5, 2321-2333 (2002).
0092-7872 1532-4125
2002
j
16W25 16N60 16R50
additive maps; higher derivations; semiprime rings; algebraic derivations; prime rings
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 \textit{V. K. Kharchenko}'s well-known theorem on algebraic derivations of prime rings [Algebra Logika 17, 220-238 (1978; Zbl 0423.16011)].
M.Brešar (Maribor)
0423.16011