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Generalized Jordan triple higher derivations on prime rings. (English) Zbl 1094.16023
Let \(R\) be a ring and both \(\{d_i\colon R\to R\mid i\geq 0\}\) and \(\{g_i\colon R\to R\mid i\geq 0\}\) be families of additive maps with \(d_0=g_0=I_R\). Consider the conditions that for all \(n\geq 0\) and all \(a,b\in R\) that: A) \(g_n(ab)=\sum_{i+j=n}g_i(a)d_j(b)\), and B) \(g_n(aba)=\sum_{i+j+k=n}g_i(a)d_j(b)d_k(a)\). The main theorem shows that if \(R\) is a prime ring with \(\text{char\,}R\neq 2\), then if B holds both as stated and also when \(g_i=d_i\) then A holds both as stated and when \(g_i=d_i\).

16W20 Automorphisms and endomorphisms
16N60 Prime and semiprime associative rings
16W25 Derivations, actions of Lie algebras