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

Yılmaz, Damla; Yazarlı, Hasret

doi:10.22075/ijnaa.2023.28801.3994

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	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
تاریخ دریافت: 02 آبان 1401، تاریخ بازنگری: 26 بهمن 1401، تاریخ پذیرش: 12 اسفند 1401
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

مراجع
[1] K.I. Beidar, W.S. III Martindale and A.V. Mikhalev, Rings with Generalized Identities, Pure and Applied Mathematics, Dekker, New York, 1996. [2] K.I. Beidar, M. Bresar and M.A. Chebotar, Functional identities with r-independent coefficients, Commun. Algebra 30 (2002), no.12, 5725–5755. [3] C.L. Chuang and T.K. Lee, Rings with annihilator conditions on multilinear polynomials, Chin J. Math. 24 (1996), no. 2, 177–185. [4] C.L. Chuang, GPI’s having coefficients in Utumi quotient rings, Proc. Am. Math. Soc. 103 (1998), no. 3, 723–728. [5] V. De Filippis, Product of two generalized derivations on polynomials in prime rings, Collectanea Mathematica 61 (2010), no. 3, 303–322. [6] C¸ . Demir and N. Arga¸c, Prime rings with generalized derivations on right ideals, Algebra Colloq. 18 (2011) no. 1, 987–998. [7] B. Dhara, b-generalized derivations on multilinear polynomials in prime rings, Bull. Korean Math. Soc. 55 (2018), no. 2, 573–586. [8] T.S. Erickson, W.S. III, Martindale and J.M. Osborn, Prime non-associative algebras, Pac. J. Math. 60 (1975), 49–63. [9] M. Fosner and J. Vukman, Identities with generalized derivations in prime rings, Mediterr J. Math. 9 (2012), no. 4, 847–863. [10] N. Jacobson, Structure of rings, American Mathematical Society, USA, 1964. [11] V.K. Kharchenko, Differential identities of prime rings, Algebra Log. 17 (1978), 155–168. [12] M.T. Ko¸san and T.K. Lee, b-Generalized derivations having nilpotent values, J. Aust. Math. Soc. 96 (2014), no. 3, 326–337. [13] C. Lanski, Differential identities of prime rings, Kharchenko’s theorem and applications, Contemp. Math. 124 (1992), 111–128. [14] T.K. Lee, Derivations with invertible values on a multilinear polynomial, Proc. Am. Math. Soc. 119 (1993), no. 4, 1077–1083. [15] U. Leron, Nil and power central polynomials in rings, Trans. Amer. Math. Soc. 202 (1975), 97–103. [16] W.S. III Martindale, Prime rings satisfying a generalized polynomial identity, J. Algebra Colloq. 3 (1996), no. 4, 369–478. [17] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100. [18] F. Rania and G. Scudo, A quadratic differential identity with generalized derivations on multilinear polynomials in prime rings, Mediterr. J. Math. 11 (2014), 273–285. [19] S.K. Tiwari and B. Prajapati Centralizing b-generalized derivations on multilinear polynomials, Filomat 33 (2019), no. 19, 6251–6266. [20] T.L. Wong, Derivations with power central values on multilinear polynomials, Algebra Colloq. 3 (1996), no. 4, 369–378. [21] T.L. Wong, Derivations cocentralizing multilinear polynomials, Taiwan J. Math. 1 (1997), no. 1, 31–37.
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A note on b-generalized derivations with a quadratic equation in prime rings