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

Document Type : Research Paper


1 Department of Mathematics, Faculty of Science, Erzurum Technical University, Erzurum, Turkey

2 Department of Mathematics, Faculty of Science, Sivas Cumhuriyet University, Sivas, Turkey


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 $R$.


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Volume 14, Issue 5
May 2023
Pages 199-209
  • Receive Date: 24 October 2022
  • Revise Date: 15 February 2023
  • Accept Date: 03 March 2023