On generalized derivations involving prime ideals with involution

Nadeem ur Rehman; Hafedh M.  Alnoghashi; Motoshi  Hongan

doi:10.3842/umzh.v75i8.7064

Nadeem ur Rehman Department of Mathematics, Aligarh Muslim University, India https://orcid.org/0000-0003-3955-7941
Hafedh M. Alnoghashi Department of Computer Science, College of Engineering and Information Technology, Amran University, Yemen
Motoshi Hongan Maniwa, Okayama, Japan

DOI: https://doi.org/10.3842/umzh.v75i8.7064

Keywords: Prime ideal, involution, generalized derivations, commutativity

Abstract

UDC 512.5

We study the structure of the quotient ${A}/{P}$, where ${A}$ is any ring with involution $*$ and ${P}$ is a prime ideal of ${A}$. With an aim to construct a ring with involution of this kind, we study the behavior of generalized derivations satisfying the algebraic identities involving prime ideals. As a consequence, currently existing results in this field are enhanced.

References

A. Mamouni, L. Oukhtite, M. Zerra, On derivations involving prime ideals and commutativity in rings, São Paulo J. Math. Sci., 1–14 (2020). DOI: https://doi.org/10.1007/s40863-020-00187-z

B. Davaz, M. A. Raza, A note on automorphisms of Lie ideals in prime rings, Math. Slovaca, 68, № 5, 1223–1229 (2018). DOI: https://doi.org/10.1515/ms-2017-0179

F. A. A. Almahdi, A. Mamouni, M. Tamekkante, A generalization of Posner's theorem on derivations in rings, Indian J. Pure and Appl. Math., 51, № 1, 187–194 (2020). DOI: https://doi.org/10.1007/s13226-020-0394-8

H. E. Mir, A. Mamouni, L. Oukhtite, Commutativity with algebraic identities involving prime ideals, Commun. Korean Math. Soc., 35, № 3, 723–731 (2020).

M. S. Khan, S. Ali, M. Ayedh, Herstein's theorem for prime ideals in rings with involution involving pair of derivations, Commun. Algebra, 50, № 6, 2592–2603 (2022). DOI: https://doi.org/10.1080/00927872.2021.2014513

N. Rehman, M. Hongan, H. M. Alnoghashi, On generalized derivations involving prime ideals, Rend. Circ. Mat. Palermo (2), 71, № 2, 601–609 (2022). DOI: https://doi.org/10.1007/s12215-021-00639-1

N. Rehman, H. M. Alnoghashi, Action of prime ideals on generalized derivations-I; arXiv preprint, arXiv:2107.06769 (2021).

N. Rehman, H. M. Alnoghashi, A. Boua, Identities in a prime ideal of a ring involving generalized derivations, Kyungpook Math. J., 61, № 4, 727–735 (2021).

N. Rehman, H. M. Alnoghashi, $T$-commuting generalized derivations on ideals and semi-prime ideal-II, Mat. Stud., 57, № 1, 98–110 (2022).

N. Rehman, H. M. Alnoghashi, $T$-commuting generalized derivations on ideals and semi-prime ideal, Bol. Soc. Parana. Mat., 1–15 (2022). DOI: https://doi.org/10.30970/ms.57.1.98-110

N. Rehman, H. M. Alnoghashi, M. Hongan, A note on generalized derivations on prime ideals, J. Algebra and Relat. Top., 10, № 1, 159–169 (2022).

N. Rehman, E. Koç Sögütcü, H. M. Alnoghashi, A generalization of Posner's theorem on generalized derivations in rings, J. Iran. Math. Soc., 3, № 1, 1–9 (2022).