Action of three $X$-generalized derivations in prime rings

Year 2025, Early Access, 1 - 29

Basudeb Dhara , Vincenzo De Filippis , S. Kar , Manami Bera

https://doi.org/10.24330/ieja.1646846

Abstract

Let $\mathfrak{R}$ be a prime ring of characteristic different from $2$, $\mathcal{Q}_r^m$ be its maximal right ring of quotients, $\mathcal{C}$ be its extended centroid and $\omega(s_1,\ldots,s_n)$ be a noncentral multilinear polynomial over $\mathcal{C}$. Suppose that $\mathcal{H}_1$, $\mathcal{H}_2$ and $\mathcal{H}_3$ are three $X$-generalized derivations on $\mathfrak{R}$. If $$\mathcal{H}_1\bigg(\mathcal{H}_2(\omega(s_1,\ldots,s_n))\omega(s_1,\ldots,s_n)\bigg)=\mathcal{H}_3(\omega(s_1,\ldots,s_n)^2)$$ for all $s_1,\ldots,s_n\in \mathfrak{R}$, then we detail all potential configurations of the maps $\mathcal{H}_1, \mathcal{H}_2$ and $\mathcal{H}_3$.

Keywords

Generalized derivation, $X$-generalized derivation, extended centroid, multilinear polynomial, prime ring

References

• N. Bera and B. Dhara, $b$-generalized skew derivations acting on multilinear polynomials in prime rings, Comm. Algebra, 53(2) (2025), 761-780.
• C.-L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103(3) (1988), 723-728.
• V. De Filippis and O. M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Comm. Algebra, 40(6) (2012), 1918-1932.
• B. Dhara, $b$-Generalized derivations on multilinear polynomials in prime rings, Bull. Korean Math. Soc., 55(2) (2018), 573-586.
• B. Dhara, Generalized derivations acting on multilinear polynomials in prime rings, Czechoslovak Math. J., 68(1) (2018), 95-119.
• B. Dhara and N. Argac, Generalized derivations acting on multilinear polynomials in prime rings and Banach algebras, Commun. Math. Stat., 4(1) (2016), 39-54.
• B. Dhara and V. De Filippis, $b$-Generalized derivations acting on multilinear polynomials in prime rings, Algebra Colloq., 25(4) (2018), 681-700.
• T. S. Erickson, W. S. Martindale, III and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math., 60(1) (1975), 49-63.
• C. Faith and Y. Utumi, On a new proof of Litoff's theorem, Acta Math. Acad. Sci. Hungar., 14 (1963), 369-371.
• C. Gupta, On $b$-generalized derivations in prime rings, Rend. Circ. Mat. Palermo, (2), 72(4) (2023), 2703-2720.
• N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloq. Pub., 37, Amer. Math. Soc., Providence, RI, 1964.
• V. K. Kharchenko, Differential identities of prime rings, Algebra Logic, 17 (1978), 155-168.
• M. T. Kosan and T. K. Lee, $b$-Generalized derivations of semiprime rings having nilpotent values, J. Aust. Math. Soc., 96(3) (2014), 326-337.
• T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica, 20(1) (1992), 27-38.
• U. Leron, Nil and power-central polynomials in rings, Trans. Amer. Math. Soc., 202 (1975), 97-103.
• C.-K. Liu, An Engel condition with b-generalized derivations, Linear Multilinear Algebra, 65(2) (2017), 300-312.
• W. S. Martindale, III, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12 (1969), 576-584.
• T. Pehlivan and E. Albas, $b$-Generalized derivations on prime rings, Ukrainian Math. J., 74(6) (2022), 953-966.
• B. Prajapati, S. K. Tiwari and C. Gupta, $b$-generalized derivations act as a multipliers on prime rings, Comm. Algebra, 50(8) (2022), 3498-3515.
• S. K. Tiwari, Identities with generalized derivations in prime rings, Rend. Circ. Mat. Palermo, (2), 71(1) (2022), 207-223.
• S. K. Tiwari and B. Prajapati, Centralizing $b$-generalized derivations on multilinear polynomials, Filomat, 33(19) (2019), 6251-6266.
• T.-L. Wong, Derivations with power central values on multilinear polynomials, Algebra Colloq., 3(4) (1996), 369-378.