On the Structural Behavior of Multiplicative (Generalized)-Derivations via d-Algebra Structures Hicham Saber, Zakia Z. Al-Amery, Radwan M. Al-Omary, Khaled Aldwoah, Ria Egami Journal of Mathematics, 2026 In the context of a d ‐algebra structure (℧, ∗, 0), this paper aims to introduce the concept of a multiplicative (generalized)‐derivation associated with a self‐map Ξ (not necessarily a derivation). Based on this concept, the operations ∧ and composition ° will be defined, and several interesting related properties will be investigated, such as regularity, d ‐ideal, kernel, d ‐subalgebra, fixed set, and ‐invariant. Moreover, we will demonstrate that by strengthening the condition of a d ‐algebra ℧ to include the property ( ℘ ∗( ℘ ∗ ϖ )) = ϖ , the collection of all multiplicative (generalized)‐derivations forms a semigroup on ℧. Furthermore, several relevant consequences and examples will be explored.
Additivity and Central Behavior of CE-Generalized Homoderivations in Associative Rings Hicham Saber, Hafedh Alnoghashi, Radwan M. Al-Omary, Khaled Aldwoah, Bakri Younis, M. I. Elashiry Journal of Mathematics, 2026 This study examines the commutativity of a ring endowed with a special class of mappings termed centrally extended generalized homoderivations. These mappings serve as an extension of several existing concepts, including homoderivations, generalized homoderivations, and left centralizers. The paper further explores distinctive structural characteristics of the center associated with such rings, emphasizing how these mappings influence their algebraic behavior and internal consistency.
On the Structure of Quotient Rings R/P via Identities with Multiplicative (Generalized) Derivations Ali Yahya Hummdi, Hafedh Alnoghashi, Radwan M. Al-omary, Rwaida A. Bahah Mathematics, 2025 This work investigates the structure of an arbitrary ring R that contains a two-sided ideal I and a prime ideal P satisfying the condition P⊊I. Our analysis centers on the consequences of several identities that involve three multiplicative (generalized) derivations, denoted by Θ1, Θ2, Θ3:R→R. These are associated with maps θ1, θ2, θ3:R→R, which are not presumed to be additive or to be derivations themselves. The study further incorporates a non-zero derivation Δ along with two arbitrary, potentially non-additive, maps Γ1, Γ2:R→R. We establish conditions under which these identities lead to significant structural properties of the ring. To underscore the importance of our assumptions, we construct an example demonstrating that the primeness hypothesis on the ideal P is indispensable for our main conclusions.
Generalized (τ, σ)-L-Derivations in Rings Hicham Saber, Zakia Z. Al-Amery, Radwan M. Al-omary, Khaled Aldwoah, Amer Alsulami, Muntasir Suhail Mathematics, 2025 Let τ and σ:X⟶X be automorphisms of an arbitrary associative ring X, and let L be a prime ideal of X. The main objective of this article is to combine the notions of generalized L-derivations and (τ,σ)-L-derivations by introducing and analyzing a novel additive mapping Π:X→X called a generalized (τ,σ)-L-derivation associated with a (τ,σ)-L-derivation π. Later, we will examine the algebraic properties of a factor ring X/L under the influence of certain algebraic expressions containing this generalized (τ,σ)-L-derivation and lying in a prime ideal L. Through our main findings, we establish certain results under different conditions. It also provides various illustrative examples to show that our primeness hypotheses in various theorems are not exaggerated.
On b-generalized derivations and commutativity of prime rings Hafedh M. Alnoghashi, Junaid Nisar, Nadeem ur Rehman, Radwan M. Al-Omary Revista Colombiana De Matematicas, 2025 Let A be a prime ring, Z(A) its center, Q its right Martindale quotient ring, C its extended centroid, ψ a non-zero b-generalized derivation of A with associated map ξ. In this article, we prove that: (i) If [ψ(x), ψ(y)] = 0 for all x, y ∈ A, then A is either commutative or there exists q ∈ Q such that ξ = ad(q), ψ(x) = -bxq, and qb = 0. (ii) If ψ(x) ◦ ψ(y) = 0 for all x, y ∈ A, then A is either commutative with char(A) = 2 or there exists q ∈ Q such that ψ(x) = -bxq and qb = 0. Additional results are established for cases involving [ξ(x), ψ(x)] = 0 or ξ(x)◦ψ(x) = 0, where char(A) = 2. Furthermore, we give some examples that show the importance of the hypotheses of our theorems.
Jordan Higher ⋆-Left (Accordingly Right)-Centralizers on Prime Rings With Involution uday Hekmat Mahmood, Salah Mehdi Salih, Radwan M. Al-omary Al Nahrain Journal of Science, 2025 n this paper we introduce the concepts higher ⋆-left (accordingly right) centralizer, Jordan higher ⋆-left (accordingly right)centralizer. We prove Any JH⋆L(AR) Con prime ring Rhas characteristic different from 2 with involution is H⋆L(AR) Con R.
Specific Identities Involving Prime Ideals with Generalized P-derivations Radwan M Al-Omary, Ali Yahya Hummdi, Zakia Zaid Al-Amery European Journal of Pure and Applied Mathematics, 2025 In this article, we will investigate the commutativity of the factor ring $\\Re/P$, where $P$ is a prime ideal of any ring $\\Re$. This investigation will be carried out using generalized $P$-derivations $\\mho$ and $\\amalg$ associated with $P$-derivations $\\chi$ and $\\propto$, respectively, that satisfy specific functional identities linking $\\Re$ to $P$. Moreover, we will discuss some related results. Finally, to reinforce the importance of our assumption regarding the primeness of $P$, we will provide some examples.
On a Quotient Ring That Satisfies Certain Identities via Generalized Reverse Derivations Nawaf L. Alsowait, Mohammed Al-Shomrani, Radwan M. Al-omary, Zakia Z. Al-Amery Mathematics, 2025 In this article, for a prime ideal ρ of an arbitrary ring ℜ, we study the commutativity of the quotient ring ℜ/ρ, whenever ℜ admits a generalized reverse derivation ϑ associated with a reverse derivation ∂ that satisfies certain identities in ρ. Additionally, we show that, for some cases, the range of the generalized reverse derivation ϑ lies in the prime ideal ρ. Moreover, we explore several consequences and special cases. Throughout, we provide examples to demonstrate that various restrictions in the assumptions of our results are essential.
On commutativity of 2-torsion free *-prime rings with generalized derivations Mathematica, 2011
Lie Ideals and Jordan Triple Derivations in Rings Motoshi Hongan, Nadeem ur Rehman, Radwan Mohammed Al-Omary Rendiconti Del Seminario Matematico Dell Universita Di Padova Mathematical Journal of the University of Padova, 2011
