b-Generalized Skew Derivations on Multilinear Polynomials
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Authors: B. PRAJAPATI
DOI: 10.46793/KgJMat2301.021P
Abstract:
Let R be a prime ring of characteristic different from 2 with the center Z(R) and F, G be b-generalized skew derivations on R. Let U be Utumi quotient ring of R with the extended centroid C and f(x1,…,xn) be a multilinear polynomial over C which is not central valued on R. Suppose that P
Z(R) such that
![[P, [F (f (r)), f(r) ]] = [G (f (r) ),f(r )],](5914c6a725ced1c8de4365674a68f95b1x.png)
- there exist λ,μ ∈ C such that F(x) = λx, G(x) = μx for all x ∈ R;
- there exist a,b ∈ U, λ,μ ∈ C such that F(x) = ax + λx + xa, G(x) = bx + μx + xb for all x ∈ R and f(x1,…,xn)2 is central valued on R.
Keywords:
b-Generalized skew derivations, multilinear polynomials, prime rings, the extended centroid, Utumi quotient ring.
References:
[1] B. Dhara, N. Argaç and E. Albas, Vanishing derivations and co-centralizing generalized derivations on multilinear polynomials in prime rings, Comm. Algebra 44 (2016), 1905–1923.
[2] B. Dhara, b-Generalized derivations on multilinear polynomials in prime rings, Bull. Korean Math. Soc. 55(2) (2018), 573–586.
[3] B. Dhara, A note on generalized skew-derivations on multilinear polynomials in prime rings, Quaest. Math. 43(2) (2020), 251–264.
[4] C. L. Chuang and T. K. Lee, Identities with a single skew derivation, J. Algebra 288(1) (2005), 59–77.
[5] C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103(3) (1988), 723–728.
[6] C. K. Liu, An Engel condition with b-generalized derivations, Linear Multilinear Algebra 65(2) (2017), 300–312.
[7] C. Faith and Y. Utumi, On a new proof of Litoff’s theorem, Acta Math. Hungar. 14 (1963), 369–371.
[8] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100.
[9] J. H. Mayne, Centralizing automorphism of prime rings, Canad. Math. Bull. 19(1) (1976), 113–115.
[10] K. I. Beidar, W. S. Martindale and V. Mikhalev, Rings with Generalized Identities, Dekker, New York, 1996.
[11] L. Carini, V. De Filippis and G. Scudo, Identities with product of generalized skew derivations on multilinear polynomials, Comm. Algebra 44(7) (2016), 3122–3138.
[12] M. Brešar, On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J. 33 (1991), 89–93.
[13] M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra 156(2) (1993), 385–394.
[14] M. T. Košan and T. K. Lee, b-Generalized derivations having nilpotent values, J. Aust. Math. Soc. 96(4) (2014), 326–337.
[15] N. Jacobson, Structure of Rings, Amer. Math. Soc., Providence, RI, 1964.
[16] N. Argaç and V. De Filippis, Actions of generalized derivations on multilinear polynomials in prime rings, Algebra Colloq. 18 (2011), 955–964.
[17] N. Divinsky, On commuting automorphisms of rings, Transactions of the Royal Society of Canada Section III 49 (1955), 19–22.
[18] R. K. Sharma, B. Dhara and C. Garg, A result concerning generalized derivations on multilinear polynomials in prime rings, Rendiconti del Circolo Matematico di Palermo Series 2 (2018), 1–17.
[19] S. K. Tiwari, Generalized derivations with multilinear polynomials in prime rings, Comm. Algebra 46(12), (2018), 5356–5372.
[20] S. K. Tiwari and B. Prajapati, Centralizing b-generalized derivations on multilinear polynomials, Filomat 33(19), (2019), 6251–6266.
[21] S. K. Tiwari and B. Prajapati, Generalized derivations act as a Jordan homomorphism on multilinear polynomials, Comm. Algebra 47(7), (2019), 2777–2797.
[22] T. S. Erickson, W. S. Martindale 3rd and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), 49–63.
[23] T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sin. 20(1) (1992), 27–38.
[24] T. K. Lee, Identities with generalized derivations, Comm. Algebra 29(10) (2001), 4437–4450.
[25] U. Leron, Nil and power central polynomials in rings, Trans. Amer. Math. Soc. 202 (1975), 297–103.
[26] V. De Filippis and O. M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Comm. Algebra 40 (2012), 1918–1932.
[27] V. De Filippis and F. Wei, Posner’s second theorem for skew derivations on multilinear polynomials on left ideals, Houston J. Math. 38(2) (2012), 373–395.
[28] V. De Filippis and F. Wei, Centralizers of X-generalized skew derivations on multilinear polynomials in prime rings, Commun. Math. Stat. 6(1) (2018), 49–71.
[29] V. K. Kharchenko, Generalized identities with automorphisms, Algebra and Logic 14(2) (1975), 132–148.
[30] W. S. Martindale 3rd, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576–584.
[31] T. L. Wong, Derivations with power central values on multilinear polynomials, Algebra Colloq. 3 (1996), 369–378.
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