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Journals / International Electronic Journal of Algebra / 2026 / Volume: 39 Issue: 39

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

Basudeb Dhara(Belda College)
Vincenzo De Filippis(University of Messina)
S. Kar(Jadavpur University)
Manami Bera(Jadavpur University)
Pages
—
Publication Date
2026-01-10
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

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 $\mathcal{H}_3$.

📌 3 Key Takeaways

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Objective & Hypothesis: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 $\mathcal{C}$.
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Key Findings & Data:Suppose that $\mathcal{H}_1$, $\mathcal{H}_2$ and $\mathcal{H}_3$ are three $X$-generalized derivations on $\mathfrak{R}$.
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Conclusion & Impact: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 $\mathcal{H}_3$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Generalized derivation · $X$-generalized derivation · extended centroid

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 $\mathcal{H}_3$.

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Cite This Article

Dhara, B., Filippis, V. D., Kar, S., Bera, M. (2026). Action of three $X$-generalized derivations in prime rings. International Electronic Journal of Algebra, Volume: 39 Issue: 39. https://doi.org/10.24330/ieja.1646846

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