Action of three $X$-generalized derivations in prime rings

Abstract

Let $\mathfrak{R}$ be a prime ring of characteristic different from $2$, $\mathcal{Q}_r^m$ be its maximal right ring of quotients, $\mathcal{C}$ be its extended centroid and $\omega(s_1,\ldots,s_n)$ be a noncentral multilinear polynomial over $\mathcal{C}$. Suppose that $\mathcal{H}_1$, $\mathcal{H}_2$ and $\mathcal{H}_3$ are three $X$-generalized derivations on $\mathfrak{R}$. If $$\mathcal{H}_1\bigg(\mathcal{H}_2(\omega(s_1,\ldots,s_n))\omega(s_1,\ldots,s_n)\bigg)=\mathcal{H}_3(\omega(s_1,\ldots,s_n)^2)$$ for all $s_1,\ldots,s_n\in \mathfrak{R}$, then we detail all potential configurations of the maps $\mathcal{H}_1, \mathcal{H}_2$ and $\mathcal{H}_3$.

Keywords

• Generalized derivation
• $X$-generalized derivation
• extended centroid
• multilinear polynomial
• prime ring

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