(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let B be a unital -ring with a 2-torsion free that contains non-trivial symmetric idempotent. For any B_1,B_2,B_3,…,B_n B, a product B_1 ° B_2=B_1B_2+B_2B_1 is called Jordan product and B_1 B_2=B_1B_2+B_2B_1^ is recognized as a skew Jordan product. Characterize mixed Jordan triple product as Q_3(B_1,B_2,B_3)=B_1 ° B_2 B_3 and mixed Jordan n-product as Q_n(B_1,B_2,…,B_n)=B_1 ° B_2 ° … B_n for all integer n≥3. The present paper deals that a mapping which is called multiplicative mixed Jordan n-derivation, Ψ: B → B satisfies Ψ(Q_n(B_1,B_2,…,B_n))=_i=1^n Q_n(B_1, …, B_i-1, Ψ(B_i), B_i+1, …,B_n) for all B_1, B_2,…, B_n B if and only if Ψ is an additive -derivation. Finally, primary outcome is applicable in various specific categories of unital -rings and -algebras including prime -rings, prime -algebras and factor von Neumann algebras.