A note on b-generalized derivations with a quadratic equation in prime rings

	A note on b-generalized derivations with a quadratic equation in prime rings
International Journal of Nonlinear Analysis and Applications
مقاله 19، دوره 14، شماره 5، مرداد 2023، صفحه 199-209 اصل مقاله (374.37 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22075/ijnaa.2023.28801.3994
نویسندگان
Damla Yılmaz^* ¹؛ Hasret Yazarlı²
¹Department of Mathematics, Faculty of Science, Erzurum Technical University, Erzurum, Turkey
²Department of Mathematics, Faculty of Science, Sivas Cumhuriyet University, Sivas, Turkey
چکیده
Let $R$ be a prime ring of characteristic different from $2$, $C$ be its extended centroid and $Q_{r}$ be its right Martindale quotient ring and $f(t_{1},...,t_{n})$ be a multilinear polynomial over $C$, which is not central valued on $R$. Assume that $F$ is a $b$-generalized derivation on $R$ and $d$ is a derivation of $R$ such that $$ F(f(s))d(f(s))+d(f(s))F(f(s))=0$$ for all $s=(s_{1},...,s_{n})\in R^{n}$. Then either $F=0$ or $d=0$, except when $d$ is an inner derivation of $R$, there exists $\lambda \in C$ such that $F(r)=\lambda r$ for all $r\in R$ and $f(t_{1},...,t_{n})^{2}$ is central valued on $R$.
کلیدواژه‌ها
b-generalized derivation؛ multilinear polynomial؛ prime ring؛ generalized polynomial identity
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