Characterization of Multiplicative Mixed Jordan-Type Derivations on Ring with Involution

Let $\mathcal{B}$ be a unital $\varnothing$-ring with a $2$-torsion free that contains non-trivial symmetric idempotent. For any $B_1,B_2,B_3,\ldots,B_n \in \mathcal{B}$, a product $B_1 \circ B_2=B_1B_2+B_2B_1$ is called Jordan product and $B_1 \bullet B_2=B_1B_2+B_2B_1^\varnothing$ is recognized as a skew Jordan product. Characterize mixed Jordan triple product as $Q_3(B_1,B_2,B_3)=B_1 \circ B_2 \bullet B_3$ and mixed Jordan $n$-product as $Q_n(B_1,B_2,\ldots,B_n)=B_1 \circ B_2 \circ \cdots \bullet B_n$ for all integer $n\geq3$. The present paper deals that a mapping which is called multiplicative mixed Jordan $n$-derivation, $\Psi$: $\mathcal{B} \rightarrow \mathcal{B}$ satisfies $\Psi(Q_n(B_1,B_2,\ldots,B_n))=\sum_{i=1}^{n} Q_n(B_1, \ldots, B_{i-1}, \Psi(B_i), B_{i+1}, \ldots,B_n)$ for all $B_1, B_2,\ldots, B_n \in \mathcal{B}$ if and only if $\Psi$ is an additive $\varnothing$-derivation. Finally, primary outcome is applicable in various specific categories of unital $\varnothing$-rings and $\varnothing$-algebras including prime $\varnothing$-rings, prime $\varnothing$-algebras and factor von Neumann algebras.