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Additivity of Jordan \(n\)-tuple derivable maps on alternative rings. (English) Zbl 1445.17007

Given an alternative ring \(R\) (i.e., any two elements generate an associative subring), consider the Jordan product \(x\circ y=xy+yx\). Then, for a natural number \(n\), a Jordan \(n\)-tuple derivation of \(R\) is defined as a map \(\delta\) satisfying
\[ \delta\bigl(a_n\circ(\cdots(a_2\circ a_1)\cdots)\bigr)=\sum_{i=1}^na_n\circ(\cdots(\delta(a_i)\circ(\cdots(a_2\circ a_1)\cdots))\cdots). \]
The paper under review is devoted to proving that, assuming that the alternative ring has neither \(2\)-torsion nor \((2^{n-1}-1)\)-torsion, and that it contains a nontrivial idempotent satisfying some technical hypotheses, any Jordan \(n\)-tuple derivation is an additive map.

MSC:

17D05 Alternative rings
17A36 Automorphisms, derivations, other operators (nonassociative rings and algebras)
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