Power centralizing semiderivations of Lie ideals in prime rings
DOI:
https://doi.org/10.5269/bspm.68857Resumen
If a semiderivation $\mathscr{F}$ with associated automorphism $\xi$ is induced on a non-central Lie ideal $\mathscr{L}$ of $\mathfrak{A}$ such that \begin{align*} \left[\mathscr{F}(\eta), \eta \right]^{n}\in\mathcal{Z(R)}, \end{align*} where $n$ is a fixed positive integer, and $\eta\in\mathcal{L}$, then it has been proven that either \begin{align*} Char(\mathfrak{A}) =0 \end{align*} or \begin{align*} Char(\mathfrak{A})>n+1, \end{align*} then $\mathfrak{A}$ satisfies a standard identity in $4$ variables usually denoted by $s_4$.Referencias
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2. Beidar, K. I., Martindale III, W. S. and Mikhalev, V., Rings with Generalized Identities, Pure and Applied Math. Dekker, New York (1996).
3. Bergen, J., Derivations in prime ring , Can. Math. Bull., 26 (3), 263-270, (1983).
4. Bresar, M., Semiderivations of prime rings, Proc. Amer. Math. Soc. 108 (4), 859–860, (1990).
5. Carini, L. and De Filippis, V., Commutators with power central values on a Lie ideal, Pacific. J. Math., 193 (2), 278-296, (1990).
6. Chuang, C. L., Differential identities with automorphisms and antiautomorphisms II, J. Algebra, 160, 130-171, (1993).
7. Chuang, C. L., GPIs having quotients in Utumi quotient rings, Proc. Amer. Math. Soc, 103 (3), 723-728, (1998).
8. Di Vincenzo, O. M., A note on k-th commutators in an associative ring, Rend. Circ. Mat. Palmero, Series II-Tomo XLVII, 106-112, (1998).
9. Erickson, T. S., Martindale III, W. S. and Osborn, J. M., Prime nonassociative algebras, Pacific J. Math. 60, 49-63, (1975).
10. Huang, S., Semiderivations with power values on Lie ideals in prime rings , Ukrainian Math. J., 65 (6), 967-971, (2013).
11. Jacobson, N., Structure of rings, Amer. Math. Soc. Colloq. Pub., 37, Amer. Math. Soc., Providence, RI, 1964.
12. Kharchenko, V. K., Generalized identities with automorphisims, Algebra and Logic, 14, 132-148, (1975).
13. Lanski, C., Differential identities, Lie ideals and Posner’s theorems , Pac. J. Math., 134 (2), 275-297, (1975).
14. Martindale III, W. S., Prime rings satisfying a generalized polynomial identity, J. Algebra, 12, 576-584, (1969).
15. Posner, E. C., Derivations in prime rings, Proc. Amer. Math. Soc., 8, 1093-1100, (1957).
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2025-05-29
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Research Articles
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Cómo citar
Rehman, N. ur, & Nisar, J. (2025). Power centralizing semiderivations of Lie ideals in prime rings (S. . Ahmad Pary, Trans.). Boletim Da Sociedade Paranaense De Matemática, 43, 1-4. https://doi.org/10.5269/bspm.68857
