Abstract:
Let RR be a prime ring of characteristic different from 2, with Utumi quotient ring UU and extended centroid CC, δδ a nonzero derivation of RR, GG a nonzero generalized derivation of RR, and f(x1,…,xn)f(x1,…,xn) a noncentral multilinear polynomial over CC. If δ(G(f(r1,…,rn))f(r1,…,rn))=0δ(G(f(r1,…,rn))f(r1,…,rn))=0 for all r1,…,rn∈Rr1,…,rn∈R, then f(x1,…,xn)2f(x1,…,xn)2 is central-valued on RR. Moreover there exists a∈Ua∈U such that G(x)=axG(x)=ax for all x∈Rx∈R and δδ is an inner derivation of RR such that δ(a)=0δ(a)=0.
Keywords:
prime ring, differential identities, generalized derivations.
Citation:
L. Carini, V. De Filippis, “Centralizers of generalized derivations on multilinear polynomials in prime rings”, Sibirsk. Mat. Zh., 53:6 (2012), 1310–1320; Siberian Math. J., 53:6 (2012), 1051–1060
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\jour Sibirsk. Mat. Zh.
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\vol 53
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Linking options:
https://www.mathnet.ru/eng/smj2384
https://www.mathnet.ru/eng/smj/v53/i6/p1310
This publication is cited in the following 27 articles:
Nripendu Bera, Basudeb Dhara, Sukhendu Kar, “Study of commutator involving X-generalized skew derivations and annihilators in prime rings”, Asian-European J. Math., 18:01 (2025)
Vincenzo De Filippis, Giovanni Scudo, Feng Wei, “On X-generalized skew derivations and their applications”, J. Algebra Appl., 23:03 (2024)
Francesco Rania, “Power central valued b-generalized skew derivations on Lie ideals”, Rend. Circ. Mat. Palermo, II. Ser, 2024
Nripendu Bera, Basudeb Dhara, “b
-generalized skew derivations acting on multilinear polynomials in prime rings”, Communications in Algebra, 2024, 1
Nripendu Bera, Basudeb Dhara, Sukhendu Kar, “A result concerning b-generalized skew derivations on multilinear polynomials in prime rings”, Communications in Algebra, 51:3 (2023), 887
Luisa Carini, Vincenzo De Filippis, Springer Proceedings in Mathematics & Statistics, 392, Algebra and Related Topics with Applications, 2022, 59
Basudeb Dhara, Sukhendu Kar, Swarup Kuila, “A Note on Generalized Derivations of Order 2 and Multilinear Polynomials in Prime Rings”, Acta Math Vietnam, 47:4 (2022), 755
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De Filippis V., “Product of Generalized Skew Derivations on Lie Ideals”, Commun. Algebr., 49:7 (2021), 2987–3009
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Das P., Dhara B., Kar S., “Generalized Derivations With Centralizing Conditions in Prime Rings”, Commun. Korean Math. Soc., 34:1 (2019), 83–93
Eroglu M.P., “Generalized Skew Derivations With Centralizer Conditions on Multilinear Polynomials”, Beitr. Algebr. Geom., 59:1 (2018), 61–76
De Filippis V., Wei F., “Centralizers of X-Generalized Skew Derivations on Multilinear Polynomials in Prime Rings”, Commun. Math. Stat., 6:1 (2018), 49–71
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