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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 $$g_i=d_i$$.

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