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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
In the present paper, we shall study the above mentioned result for generalized derivations in rings with involution.
📌 3 Key Takeaways
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Objective & Hypothesis:Let $R$ be an associative ring.
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Key Findings & Data:In [7], Herstein proved the following result: If $R$ is a prime ring of $char(R)\neq 2$ admitting a nonzero derivation $d$ such that $[d(x),d(y)]=0$ for all $x,y\in R$, then $R$ is commutative.
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Conclusion & Impact:In the present paper, we shall study the above mentioned result for generalized derivations in rings with involution.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Prime ring · involution · derivation
Abstract
Let $R$ be an associative ring. An additive map $F:R\toR$ is called a generalized derivation if there exists a derivation $d$ of $R$ such that $F(xy)=F(x)y+xd(y)$ for all $x,y\in R$. In [7], Herstein proved the following result: If $R$ is a prime ring of $char(R)\neq 2$ admitting a nonzero derivation $d$ such that $[d(x),d(y)]=0$ for all $x,y\in R$, then $R$ is commutative. In the present paper, we shall study the above mentioned result for generalized derivations in rings with involution.
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Cite This Article
Ali, S., Khan, A. N., Dar, N. A. (2017). Herstein’s theorem for generalized derivations in rings with involution. Hacettepe Journal of Mathematics and Statistics, Volume: 46 Issue: 6. https://ajindex.com/articles/1402983