More specifically, we show that if $F$ is generalized skew-derivation of a prime ring $R$ with associated automorphism $\alpha$ such that the relation $$F(X^2)=F(X)^2+F(X)X+XF(X)$$ holds for all $X\in f(R)$, where $f(x_1,\ldots,x_n)$ is a noncentral valued multilinear polynomial over extended centroid $C$, then either $F=0$ or $F=-id_{R}$ or $F=-id_{R}+\alpha$ (where $id_{R}$ denotes the identity map of $R$).
📌 3 Key Takeaways
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Objective & Hypothesis:In this article, we are intended to examine generalized skew-derivations that act as Jordan homoderivations on multilinear polynomials in prime rings.
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Key Findings & Data:In this article, we are intended to examine generalized skew-derivations that act as Jordan homoderivations on multilinear polynomials in prime rings.
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Conclusion & Impact:More specifically, we show that if $F$ is generalized skew-derivation of a prime ring $R$ with associated automorphism $\alpha$ such that the relation $$F(X^2)=F(X)^2+F(X)X+XF(X)$$ holds for all $X\in f(R)$, where $f(x_1,\ldots,x_n)$ is a noncentral valued multilinear polynomial over extended centroid $C$, then either $F=0$ or $F=-id_{R}$ or $F=-id_{R}+\alpha$ (where $id_{R}$ denotes the identity map of $R$).
In this article, we are intended to examine generalized skew-derivations that act as Jordan homoderivations on multilinear polynomials in prime rings. More specifically, we show that if $F$ is generalized skew-derivation of a prime ring $R$ with associated automorphism $\alpha$ such that the relation $$F(X^2)=F(X)^2+F(X)X+XF(X)$$ holds for all $X\in f(R)$, where $f(x_1,\ldots,x_n)$ is a noncentral valued multilinear polynomial over extended centroid $C$, then either $F=0$ or $F=-id_{R}$ or $F=-id_{R}+\alpha$ (where $id_{R}$ denotes the identity map of $R$).
Dhara, B., Sandhu, G. S., Bera, N. (2025). A classification of generalized skew-derivations on multilinear polynomials in prime rings. Hacettepe Journal of Mathematics and Statistics, Volume: 54 Issue: 3. https://doi.org/10.15672/hujms.1472753
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