Power centralizing semiderivations of Lie ideals in prime rings
DOI:
https://doi.org/10.5269/bspm.68857Abstract
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 $s_4$.References
1. Lions, J. L., Exact Controllability, Stabilizability, and Perturbations for Distributed Systems, Siam Rev. 30, 1-68, (1988).
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).
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).
Downloads
Published
2025-05-29
Issue
Section
Research Articles
License
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
How to Cite
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
