$\mathscr{T}$-Commuting Generalized Derivations on Ideals and Semi-Prime Ideal-II

Keywords: Semi-prime ideal; generalized derivations; commutativity.


The study's primary purpose is to investigate the $\mathscr{A}/\mathscr{T}$ structure of a quotient ring, where $\mathscr{A}$ is an arbitrary ring and $\mathscr{T}$ is a semi-prime ideal of $\mathscr{A}$. In more details, we look at the differential identities in a semi-prime ideal of an arbitrary ring using $\mathscr{T}$-commuting generalized derivation. The article proves a number of statements. A characteristic representative of these assertions is, for example, the following Theorem 3: Let $\mathscr{A}$ be a ring with $\mathscr{T}$ a semi-prime ideal and $\mathscr{I}$ an ideal of $\mathscr{A}.$ If $(\lambda, \psi)$ is a non-zero generalized derivation of $\mathscr{A}$ and the derivation satisfies any one of the conditions:\
1)\ $\lambda([a, b])\pm[a, \psi(b)]\in \mathscr{T}$,\ 2) $\lambda(a\circ b)\pm a\circ \psi(b)\in \mathscr{T}$,
$\forall$ $a, b\in \mathscr{I},$ then $\psi$ is $\mathscr{T}$-commuting on $\mathscr{I}.$

Furthermore, examples are provided to demonstrate that the constraints placed on the hypothesis of the various theorems were not unnecessary.

Author Biographies

N. U. Rehman, Aligarh Muslim University

Aligarh Muslim University

H. M. Alnoghashi, Aligarh Muslim University

Aligarh Muslim University


F.A.A. Almahdi, A. Mamouni, M. Tamekkante, A generalization of Posner’s theorem on derivations in rings, Indian J. Pure Appl. Math., 51 (2020), №1, 187–194.

M. Ashraf, A. Ali, S. Ali, Some commtativity theorems for rings with generalized derivations, Southeast Asian Bull. Math., 31 (2007), 415–421.

M. Ashraf, N. Rehman, On commutativity of rings with dervations, Results Math., 42 (2002), №1–2,3–8.

H.E. Bell, W.S. Martindale III, Centralizing mappings of semiprime rings, Canad. Math. Bull., 30 (1987), №1, 92–101.

M. Hongan, A note on semiprime rings with derivations, Internat. J. Math. Math. Sci., 20 (1997), №2, 413–415.

C. Lanski, Differential identities, Lie ideals and Posner’s theorems, Pacific J. Math., 134 (1988), №2, 275–297.

J. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull., 19 (1976), 113–115.

H.El Mir, A. Mamouni, L. Oukhtite, Commutativity with algebraic identities involving prime ideals, Communications of the Korean Mathematical Society, 35, №3, (2020), 723–731.

M.A. Idrissi, L. Oukhtite, Structure of a quotient ring R/P with generalized derivations acting on prime ideal P and some applications, Indian J. Pure Appl. Math. doi: 10.1007/s13226-021-00173-x

A. Mamouni, B. Nejjar, L. Oukhtite, Differential identities on prime rings with involution, J. Algebra & Appl., 17 (2018), №9, 11 p.

A. Mamouni, L. Oukhtite, M., Zerra, On derivations involving prime ideals and commutativity in rings, S˜ao Paulo Journal of Mathematical Sciences, 14 (2020), 675–688.

A. Mamouni, L. Oukhtite, M. Zerra, Prime ideals and generalized derivations with central values on rings, Rendiconti del Circolo Matematico di Palermo Series 2. doi: 10.1007/s12215-020-00578-3

H. Nabiel, Ring subsets that be center-like subsets, J. Algebra Appl., 17 (2018), №3, 8 p.

E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), №6, 1093–1100.

M.A. Quadri, M.S. Khan, N. Rehman, Generalized derivations and commutativity of prime rings, Indian J. Pure Appl. Math., 34 (2003), №9, 1393–1396.

S.K. Tiwari, R.K. Sharma, B. Dhara, Identities related to generalized derivation on ideal in prime rings, Beitr. Algebra Geom., 57 (2016), №4, 809–821.

N. Rehman, M.A. Raza, On m-commuting mappings with skew derivations in prime rings, St. Petersburg Math. J., 27 (2016), 641–650.

N. Rehman, On Lie ideals and automorphisms in prime rings, Math. Notes, 107 (2020), №1, 140–144.

N. Rehman, E.K. S¨og¨utc¨u, H.M. Alnoghashi, A generalization of Posner’s theorem on generalized derivations in rings, preprint.

N. Rehman, M. Hongan, H.M. Alnoghashi, On generalized derivations involving prime ideals, Rendiconti del Circolo Matematico di Palermo Series 2. doi: 10.1007/s12215-021-00639-1

How to Cite
Rehman, N. U., & Alnoghashi, H. M. (2022). $\mathscr{T}$-Commuting Generalized Derivations on Ideals and Semi-Prime Ideal-II. Matematychni Studii, 57(1), 98-110. https://doi.org/10.30970/ms.57.1.98-110