Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-682214520230501A note on b-generalized derivations with a quadratic equation in prime rings199209752810.22075/ijnaa.2023.28801.3994ENDamla YılmazDepartment of Mathematics, Faculty of Science, Erzurum Technical University, Erzurum, Turkey0000-0002-6741-8669Hasret YazarlıDepartment of Mathematics, Faculty of Science, Sivas Cumhuriyet University, Sivas, TurkeyJournal Article20221024Let $R$ 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 $$ F(f(s))d(f(s))+d(f(s))F(f(s))=0$$<br />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 $R$.https://ijnaa.semnan.ac.ir/article_7528_7ec9b0296ee16f31d4df1ca6067aeab2.pdf