Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2138 https://internationalpubls.com SCP Identities of Multiplicative (Generalized)-Derivations on One-Sided Ideals in Semiprime Rings Ahmed Aboubakr1,2* and Nawal M. NourEldeen1,3 1Department of Mathematics, Collage of Science, Taibah University, Medina, Saudi Arabia. 2Department of Mathematics, Faculty of Science, Fayoum University, 63514 Fayoum, Egypt. aaboubakr@taibahu.edu.sa & afs00@fayoum.edu.eg 3Department of Mathematics, Women’s College of Arts, Sciences and Education , Ain Shams University, Egypt. neldeen@taibahu.edu.sa Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: Let ℬ be a ring. A map Γ: ℬ → ℬ is termed a multiplicative (generalized)-derivation (abbreviated as 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 throughout this paper) if Γ(𝜈𝜔) = Γ(𝜈)𝜔 + 𝜈𝛿(𝜔) holds ∀ 𝜈,𝜔 ∈ ℬ where 𝛿: ℬ → ℬ is any map (not necessarily a derivation). If Γ, Δ are 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 associated with maps 𝛿, 𝜉 respectively. This paper aims to investigate the following algebric identities: (i) [Γ(𝜈), Γ(𝜔)] = ±[𝜈,𝜔] (SCP map Γ), (ii)[Γ(𝜈),𝜔] = ±[𝜈, Δ(𝜔)], (iii) [𝜉(𝜈), Γ(𝜔)] = ±[𝜈,𝜔] and (iv) [𝜉(𝜈), 𝜔] = ±[𝜈, Γ(𝜔)] for all 𝜈,𝜔 in a left ideal of a semiprime ring ℬ. Additionally, we present an example illustrating that the semiprimeness condition in our theorems cannot be omitted. Mathematics Subject Classification (MSC2020): 16W25, 16N60, 16U80. Keywords: Semiprime rings, l-ideals, generalized derivation, multiplicative (generalized)- derivation. 1- Introduction The concept of multiplicative derivation was first established in 1991 by Daif [1] who defined it as: A function 𝛿: ℬ → ℬ (not necessarily additive) is termed a multiplicative derivation of ℬ if 𝛿(𝜈𝜔) = 𝛿(𝜈)𝜔 + 𝜈𝛿(𝜔) for all 𝜈,𝜔 ∈ ℬ. The research by Daif [1] drew inspiration from Martindale’s work [2]. Subsequently, Goldmann and Ŝemrl provided a thorough characterization of these functions in [3]. This initial definition was later broadened to multiplicative generalized derivation by Daif and Tammam in [4], who characterized it as: A function Γ: 𝑅 → ℬ (not necessarily additive) qualifies as a multiplicative generalized derivation if there exists a multiplicative derivation 𝛿: ℬ → ℬ such that Γ(𝜈𝜔) = Γ(𝜈)𝜔 + 𝜈𝛿(𝜔) ∀ 𝜈,𝜔 ∈ ℬ. Later, Dhara and Ali [5] expanded this definition of multiplicative generalized derivation by allowing 𝛿 to be any function on ℬ. Therefore, a function Γ: 𝑅 → ℬ (not necessarily additive) is designated a 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 (𝑀𝑢𝑙𝑡. (𝐺) − 𝐷) if Γ(𝜈𝜔) = Γ(𝜈)𝜔 + 𝜈𝛿(𝜔) is satisfied ∀ 𝜈, 𝜔 ∈ ℬ, where 𝛿: 𝑅 → ℬ represents any function (not necessarily a derivation nor additive). Consequently, the framework of 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 encompasses the framework of multiplicative derivation. Additionally, 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 with 𝛿 = 0 encompasses the notion of multiplicative centralizer (not necessarily additive). mailto:afs00@fayoum.edu.eg Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2139 https://internationalpubls.com It is evident that every generalized derivation constitutes a 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 on ℬ. Nonetheless, the converse does not necessarily hold in all instances, as illustrated by the subsequent examples: Example 1.1 Consider ℬ = 𝐶[0,1], the ring of all continuous real functions and define maps 𝛿: ℬ → ℬ, as follows: 𝛿(𝑓)(𝜈) = {0, otherwise. 𝑓(𝜈)𝑙𝑜𝑔|𝑓(𝜈)| ∀ 𝑓(𝜈)≠0 And Γ: ℬ → ℬ, as follows, Γ(𝑓)(𝜈) = {0, otherwise. 𝑓(𝜈)(1+𝑙𝑜𝑔|𝑓(𝜈)|) ∀ 𝑓(𝜈)≠0 It is evident that 𝛿 and Γ do not exhibit additive properties, while 𝐷 functions as a multiplicative derivation, and Γ is classified as a 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 associated with 𝐷. Example 1.2 Consider the ring ℬ = {, ( 0 0 𝑏 𝑐 0 0 0 𝑑 0 0 0 0 0 0 0 0 ) |𝑏, 𝑑, 𝑐 ∈ ℝ}. Define maps 𝛤: ℬ → ℬ and 𝛿:ℬ → ℬ as follows: 𝛤( ( 0 0 𝑏 𝑐 0 0 0 𝑑 0 0 0 0 0 0 0 0 ) ) = ( 0 0 0 𝑏𝑑 0 0 0 0 0 0 0 0 0 0 0 0 ) , and 𝛿( ( 0 0 𝑏 𝑐 0 0 0 𝑑 0 0 0 0 0 0 0 0 ) ) = ( 0 0 0 𝑐2 0 0 0 0 0 0 0 0 0 0 0 0 ) . It is evident that 𝛿 and Γ do not exhibit additive properties, while 𝐷 functions as a multiplicative derivation, and Γ is classified as a 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 associated with 𝐷. In the last thirty years, some researchers have demonstrated commutativity theorems for certain prime or semiprime rings having automorphisms or derivations which are centralizing or commuting on certain ordered subsets of ℬ. (see [6], [7], [8], [9], [10] and [11], where further references can be found). Let 𝑆 be a non-empty subset of ℬ. We define a function Γ:ℬ → ℬ as commutativity preserving on a subset 𝑆 of ℬ when the following condition holds: for any 𝜈, 𝜔 ∈ 𝑆, if [𝜈, 𝜔] = 0, then [Γ(𝜈), Γ(𝜔)] = 0. Furthermore, the function Γ is termed strong commutativity preserving (abbreviated as SCP) on 𝑆 if ∀ elements 𝜈,𝜔 ∈ 𝑆, the equality [𝜈, 𝜔] = [Γ(𝜈), Γ(𝜔)] is satisfied. An expanding body of research exists concerning strong commutativity preserving (SCP) functions and derivations (see works by [12], [13], [14], among others) In [15], Ali demonstrated that when ℬ represents a semiprime ring and 𝑓 constitutes an endomorphism that functions as a SCP map on a nonzero ideal 𝑈 of ℬ, then 𝑓 necessarily operates as a commuting map on 𝑈. Additionally, in [16], Samman established that an epimorphism of a semiprime ring exhibits Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2140 https://internationalpubls.com strong commutativity preserving properties if and only if it is centralizing. Both derivations and SCP maps have been thoroughly investigated by numerous scholars within the domains of operator algebras, prime rings, and semiprime rings as well. This article aims to establish several theorems concerning 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 on semiprime rings that hold significant independent value. Specifically, our investigations extend previously established results by generalizing in two directions: first, by replacing a two-sided ideal with a left-sided ideal 𝐿 (abbreviated as l-ideal), and second, by substituting a generalized derivation with a 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 within the framework of semiprime rings. For our analysis, we will examine Γ and Δ as 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 associated with the maps 𝛿 and 𝜉 respectively. Our research will focus on investigating the following algebraic conditions: [Γ(𝜈), Γ(𝜔)] = ±[𝜈,𝜔] (SCP map Γ), [Γ(𝜈), 𝜔] = ±[𝜈, Δ(𝜔)], [𝜉(𝜈), Γ(𝜔)] = ±[𝜈,𝜔] and [𝜉(𝜈), 𝜔] = ±[𝜈, Γ(𝜔)] ∀𝜈,𝜔 ∈ 𝐿. The following lemmas will be essential for establishing our results. Lemma 1.3 [17] . Let ℬ be a 2-torsion free semiprime ring and 𝐿 a l-ideal of ℬ. If elements 𝑎, 𝑏 ∈ ℬ satisfy the condition 𝑎𝜈𝑏 + 𝑏𝜈𝑎 = 0 ∀𝜈 ∈ 𝐿, then 𝑎𝜈𝑏 = 0 and 𝑏𝜈𝑎 = 0 ∀𝜈 ∈ 𝐿. Lemma 1.4 [11, Lemma 2.1] . Let ℬ be a semiprime ring, ℒ a nonzero two-sided ideal of ℬ. If an element 𝑎 ∈ ℒ satisfies the condition 𝑎𝜈𝑎 = 0 ∀𝜈 ∈ ℒ, then 𝑎 = 0. 2- SCP condition on 𝑴𝒖𝒍𝒕. (𝑮) − 𝑫 on l-ideal Theorem 2.1 Consider a 2-torsion free semiprime ring ℬ, 𝐿 a nonzero l-ideal of ℬ and 𝛤:ℬ → ℬ a 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 linked to the map 𝜉. If 𝛤(𝜈𝜔) = 𝜈𝛤(𝜔) + 𝜉(𝜈)𝜔 𝜈,𝜔 ∈ 𝐿 and 𝛤 is SCP on 𝐿, then 𝐿[𝜉(𝜈), 𝜈] = 0 and 𝐿[𝛤(𝜈), 𝜈] = 0, ∀𝜈 ∈ 𝐿. Proof. Given that Γ exhibits the SCP property on 𝐿, it follows that [Γ(𝜈), Γ(𝜔)] = [𝜈, 𝜔] ∀ 𝜈, 𝜔 ∈ 𝐿. (1) By substituting 𝜔 with 𝜔𝜈 in (1), we obtain [Γ(𝜈), Γ(𝜔)]𝜈 + Γ(𝜔)[Γ(𝜈), 𝜈] + [Γ(𝜈), 𝜔]𝜉(𝜈) + 𝜔[Γ(𝜈), 𝜉(𝜈)] = [𝜈, 𝜔]𝜈 ∀ 𝜈, 𝜔 ∈ 𝐿. (2) Multiplying (1) to the right by 𝜈, we have [Γ(𝜈), Γ(𝜔)]𝜈 = [𝜈, 𝜔]𝜈 ∀ 𝜈, 𝜔 ∈ 𝐿. (3) Combining (2) and (3), we obtain Γ(𝜔)[Γ(𝜈), 𝜈] + [Γ(𝜈), 𝜔]𝜉(𝜈) + 𝜔[Γ(𝜈), 𝜉(𝜈)] = 0 ∀ 𝜈, 𝜔 ∈ 𝐿. (4) Now, if we replace 𝜔 with 𝑧𝜔 in (4) and apply (4), we obtain 𝜉(𝑧)𝜔[Γ(𝜈), 𝜈] + [Γ(𝜈), 𝑧]𝜔𝜉(𝜈) = 0 ∀ 𝜈, 𝜔, 𝑧 ∈ 𝐿. (5) Take 𝑧 = 𝜈, we have 𝜉(𝜈)𝜔[Γ(𝜈), 𝜈] + [Γ(𝜈), 𝜈]𝜔𝜉(𝜈) = 0 ∀𝜈, 𝜔 ∈ 𝐿. Lemma 1.3, gives [Γ(𝜈), 𝜈]𝜔𝜉(𝜈) = 0 for all 𝜈, 𝜔 ∈ 𝐿. (6) Since Γ(𝜈2) = Γ(𝜈)𝜈 + 𝜈𝜉(𝜈) = 𝜈Γ(𝜈) + 𝜉(𝜈)𝜈 ∀𝜈 ∈ 𝐿, this gives [Γ(𝜈), 𝜈] = [𝜉(𝜈), 𝜈] ∀ 𝜈 ∈ 𝐿. (7) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2141 https://internationalpubls.com Using (7) in (6) we have [𝜉(𝜈), 𝜈]𝜔𝜉(𝜈) = 0 ∀𝜈 ∈ 𝐿, that is 𝐿[𝜉(𝜈), 𝜈]ℬ𝐿[𝜉(𝜈), 𝜈] = 0 ∀𝜈 ∈ 𝐿. Using the semiprimeness of ℬ, we deduce that 𝐿[𝜉(𝜈), 𝜈] = 0 ∀ 𝜈 ∈ 𝐿. Substituting into (7), we obtain 𝐿[Γ(𝜈), 𝜈] = 0 ∀𝜈 ∈ 𝐿. The case [Γ(𝜈), Γ(𝜔)] = −[𝜈, 𝜔] ∀𝜈,𝜔 ∈ 𝐿 is similar. Remark 2.2 The result can be established in a similar manner for the case where [𝛤(𝜈), 𝛤(𝜔)] = −[𝜈,𝜔] holds ∀𝜈, 𝜔 ∈ 𝐿. Corollary 2.3 Let ℬ be a 2-torsion free semiprime ring, and let 𝛤: ℬ → ℬ be a 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 associated with the map 𝜉. If 𝛤(𝜈𝜔) = 𝜈𝛤(𝜔) + 𝜉(𝜈)𝜔 ∀𝜈,𝜔 ∈ ℬ, and if 𝛤 is SCP on 𝐿 or satisfies [𝛤(𝜈), 𝛤(𝜔)] = −[𝜈,𝜔] ∀𝜈,𝜔 ∈ ℬ, then 𝜉 and 𝛤 commute on ℬ. Theorem 2.4 Let ℬ be a semiprime ring, 𝐿 a nonzero left ideal of ℬ, and 𝛤, 𝛥 two 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 associated with the maps 𝛿 and 𝜉, respectively. If [𝛤(𝜈), 𝜔] = [𝜈, 𝛥(𝜔)] or [𝛤(𝜈), 𝜔] = −[𝜈, 𝛥(𝜔)], ∀𝜈, 𝜔 ∈ 𝐿, then 𝐿[𝛿(𝜈), 𝜈] = 0 and 𝐿[𝜉(𝜈), 𝜈] = 0, ∀𝜈 ∈ 𝐿. Proof. Assume [Γ(𝜈), 𝜔] = [𝜈, Δ(𝜔)] ∀ 𝜈, 𝜔 ∈ 𝐿. (8) If we replace in (8) 𝜈 by 𝜈𝜔, we get [Γ(𝜈), 𝜔]𝜔 + [𝜈,𝜔]𝛿(𝜔) + 𝜈[𝛿(𝜔),𝜔] = [𝜈, Δ(𝜔)]𝜔 + 𝜈[𝜔, Δ(𝜔)] ∀ 𝜈, 𝜔 ∈ 𝐿. (9) On the other hand, if we multiply (8) on the right by 𝜔, we obtain [Γ(𝜈), 𝜔]𝜔 = [𝜈, Δ(𝜔)]𝜔 ∀ 𝜈, 𝜔 ∈ 𝐿. (10) Subtracting (10) from (9), we get [𝜈, 𝜔]𝛿(𝜔) + 𝜈[𝛿(𝜔), 𝜔] = 𝜈[𝜔, Δ(𝜔)], ∀𝜈, 𝜔 ∈ 𝐿. (11) Now if we substitute 𝜈 by 𝑟𝜈 in (11), we have 𝑟[𝜈, 𝜔]𝛿(𝜔) + [𝑟, 𝜔]𝜈𝛿(𝜔) + 𝑟𝜈[𝛿(𝜔),𝜔] = 𝑟𝜈[𝜔, Δ(𝜔)] ∀ 𝜈, 𝜔 ∈ 𝐿, 𝑟 ∈ ℬ. (12) If we multiply (11) on the left by 𝑟, we get 𝑟[𝜈, 𝜔]𝛿(𝜔) + 𝑟𝜈[𝛿(𝜔),𝜔] = 𝑟𝜈[𝜔, Δ(𝜔)] ∀ 𝜈, 𝜔 ∈ 𝐿, 𝑟 ∈ ℬ. (13) From (12) and (13), we obtain [𝑟, 𝜔]𝜈𝛿(𝜔) = 0. If we substitute 𝑟 with 𝛿(𝜔), we have [𝛿(𝜔), 𝜔]𝜈𝛿(𝜔) = 0 ∀ 𝜈, 𝜔 ∈ 𝐿. (14) If we multiply, (14) to the right by 𝜔, we get [𝛿(𝜔), 𝜔]𝜈𝛿(𝜔)𝜔 = 0. Replacing 𝜈 by 𝜈𝜔 in (14), to get [𝛿(𝜔),𝜔]𝜈𝜔𝛿(𝜔) = 0. By subtracting the last two identities, we arrive at [𝛿(𝜔), 𝜔]𝜈[𝛿(𝜔),𝜔] = 0. Since 𝐿 is a left ideal, it follows that 𝐿[𝛿(𝜔), 𝜔]ℬ𝐿[𝛿(𝜔),𝜔] = 0 ∀ 𝜔 ∈ 𝐿. Using the semiprimeness of ℬ, we conclude that 𝐿[𝛿(𝜔),𝜔] = 0, ∀𝜔 ∈ 𝐿. On the other hand, if we replace 𝜔 with 𝜔𝜈 in (8), we obtain [Γ(𝜈), 𝜔]𝜈 + 𝜔[Γ(𝜈), 𝜈] = [𝜈, Δ(𝜔)]𝜈 + 𝜔[𝜈, 𝜉(𝜈)] + [𝜈, 𝜔]𝜉(𝜈) ∀ 𝜈, 𝜔 ∈ 𝐿. (15) Using (8) in (15) we get 𝜔[Γ(𝜈), 𝜈] = 𝜔[𝜈, 𝜉(𝜈)] + [𝜈, 𝜔]𝜉(𝜈) ∀ 𝜈, 𝜔 ∈ 𝐿. (16) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2142 https://internationalpubls.com We can proceed in a similar manner as above and deduce that 𝐿[𝜉(𝜔), 𝜔] = 0, ∀𝜔 ∈ 𝐿. The case where [Γ(𝜈), 𝜔] = −[𝜈, Δ(𝜔)], ∀𝜈, 𝜔 ∈ 𝐿 follows analogously. Corollary 2.5 Let ℬ be a semiprime ring, ℒ a nonzero two-sided ideal, and 𝛤, 𝛥 two 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 associated with the maps 𝛿 and 𝜉, respectively. If [𝛤(𝜈), 𝜔] = [𝜈, 𝛥(𝜔)] or [𝛤(𝜈), 𝜔] = −[𝜈, 𝛥(𝜔)]∀𝜈, 𝜔 ∈ ℒ, then 𝛿 and 𝜉 commute on ℒ. Proof. By Theorem 2.4, we have ℒ[𝛿(𝜔),𝜔] = 0∀𝜔 ∈ ℒ. Multiplying on the left by [𝛿(𝜔), 𝜔], we obtain [𝛿(𝜔), 𝜔]ℒ[𝛿(𝜔),𝜔] = 0. By Lemma 1.4, it follows that [𝛿(𝜔), 𝜔] = 0∀𝜔 ∈ ℒ, which implies that 𝛿 commutes on ℒ. Similarly, we can show that 𝜉 also commutes on ℒ. Theorem 2.6 Let ℬ represent a semiprime ring that is free of 2-torsion, 𝐿 a nonzero l-ideal within ℬ, and 𝛤: ℬ → ℬ a 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 mapping linked to 𝜉. Suppose that ∀𝜈, 𝜔 ∈ 𝐿, the relation 𝛤(𝜈𝜔) = 𝜈𝛤(𝜔) + 𝜉(𝜈)𝜔 holds, along with either [𝜉(𝜈), 𝛤(𝜔)] = [𝜈, 𝜔] or [𝜉(𝜈), 𝛤(𝜔)] = −[𝜈, 𝜔]. Then 𝐿[𝜉(𝜈), 𝜈] = 0 and 𝐿[𝛤(𝜈), 𝜈] = 0, ∀𝜈 ∈ 𝐿. Proof. Assume that [𝜉(𝜈), Γ(𝜔)] = [𝜈, 𝜔] ∀ 𝜈, 𝜔 ∈ 𝐿. (17) Substituting 𝜔 with 𝜔𝜈 in (17), we obtain [𝜉(𝜈), Γ(𝜔)]𝜈 + Γ(𝜔)[𝜉(𝜈), 𝜈] + [𝜉(𝜈), 𝜔]𝜉(𝜈) = [𝜈, 𝜔]𝜈 ∀ 𝜈, 𝜔 ∈ 𝐿. (18) Multiplying (17) to the right by 𝜈, we obtain [𝜉(𝜈), Γ(𝜔)]𝜈 = [𝜈, 𝜔]𝜈 ∀ 𝜈, 𝜔 ∈ 𝐿. (19) Combining (18) and (19) we obtain Γ(𝜔)[𝜉(𝜈), 𝜈] + [𝜉(𝜈), 𝜔]𝜉(𝜈) = 0 ∀ 𝜈, 𝜔 ∈ 𝐿. (20) If we substitute 𝑧𝜔 for 𝜔 in (20), the result is 𝑧Γ(𝜔)[𝜉(𝜈), 𝜈] + 𝜉(𝑧)𝜔[𝜉(𝜈), 𝜈] + 𝑧[𝜉(𝜈), 𝜔]𝜉(𝜈) + [𝜉(𝜈), 𝑧]𝜔𝜉(𝜈) = 0 ∀ 𝜈, 𝜔, 𝑧 ∈ 𝐿. (21) Using (20) in (21) and take 𝑧 = 𝜈, we get 𝜉(𝜈)𝜔[𝜉(𝜈), 𝜈] + [𝜉(𝜈), 𝜈]𝜔𝜉(𝜈) = 0 ∀ 𝜈, 𝜔 ∈ 𝐿. (22) Lemma 1.3, gives [𝜉(𝜈), 𝜈]𝜔𝜉(𝜈) = 0, ∀𝜈, 𝜔 ∈ 𝐿, this gives 𝐿[𝜉(𝜈), 𝜈]ℬ𝐿[𝜉(𝜈), 𝜈] = 0, ∀𝜈 ∈ 𝐿. Given that ℬ is semiprime, it follows that 𝐿[𝜉(𝜈), 𝜈] = 0, ∀𝜈 ∈ 𝐿. Since Γ(𝜈2) = Γ(𝜈)𝜈 + 𝜈𝜉(𝜈) = 𝜈Γ(𝜈) + 𝜉(𝜈)𝜈 ∀ 𝜈 ∈ 𝐿, this gives [Γ(𝜈), 𝜈] = [𝜉(𝜈), 𝜈], thus 𝐿[Γ(𝜈), 𝜈] = 𝐿[𝜉(𝜈), 𝜈] = 0, ∀𝜈 ∈ 𝐿. The case [𝜉(𝜈), Γ(𝜔)] = −[𝜈, 𝜔], ∀𝜈, 𝜔 ∈ 𝐿 is similar. Corollary 2.7 Let ℬ be a semiprime ring that is free of 2-torsion, and let 𝛤: ℬ → ℬ be a 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 mapping associated with 𝜉. If 𝛤(𝜈𝜔) = 𝜈𝛤(𝜔) + 𝜉(𝜈)𝜔 and [𝜉(𝜈), 𝛤(𝜔)] = [𝜈, 𝜔] or [𝜉(𝜈), 𝛤(𝜔)] = −[𝜈,𝜔] holds ∀𝜈,𝜔 ∈ ℬ, then 𝜉 and 𝛤 commute on ℬ. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2143 https://internationalpubls.com Theorem 2.8 Let ℬ represent a semiprime ring, 𝐿 a nonzero l-ideal of ℬ, and 𝛤 a 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 linked to a mapping 𝜉. If it holds that [𝜉(𝜈), 𝜔] = [𝜈, 𝛤(𝜔)] or [𝜉(𝜈), 𝜔] = −[𝜈, 𝛤(𝜔)] ∀ elements 𝜈, 𝜔 ∈ 𝐿, then it follows that 𝐿[𝛿(𝜈), 𝜈] = 0 and 𝐿[𝛤(𝜈), 𝜈] = 0, ∀𝜈 ∈ 𝐿. Proof. Assume that [𝜉(𝜈), 𝜔] = [𝜈, Γ(𝜔)] ∀ 𝜈, 𝜔 ∈ 𝐿. (23) By substituting 𝜔 with 𝜔𝜈 in (23), we derive [𝜉(𝜈), 𝜔]𝜈 + 𝜔[𝜉(𝜈), 𝜈] = [𝜈, Γ(𝜔)]𝜈 + 𝜔[𝜈, 𝜉(𝜈)] + [𝜈, 𝜔]𝜉(𝜈) ∀ 𝜈, 𝜔 ∈ 𝐿. (24) Multiplying (23) to the right by 𝜈, we obtain [𝜉(𝜈), 𝜔]𝜈 = [𝜈, Γ(𝜔)]𝜈 ∀ 𝜈, 𝜔 ∈ 𝐿. (25) Combining (24) and (25), we have 2𝜔[𝜉(𝜈), 𝜈] = [𝜈, 𝜔]𝜉(𝜈) ∀ 𝜈, 𝜔 ∈ 𝐿. (26) Now, by substituting 𝜔 with 𝑧𝜔 in (26) and applying (26), we obtain [𝜈, 𝑧]𝜔𝜉(𝜈) = 0 ∀ 𝜈, 𝜔, 𝑧 ∈ 𝐿. (27) By replacing 𝑧 with 𝑟𝑧 in (27), we obtain [𝜈, 𝑟]𝑧𝜔𝜉(𝜈) = 0 ∀ 𝜈, 𝜔, 𝑧 ∈ 𝐿𝑟 ∈ ℬ. (28) Replacing 𝑧 by 𝜉(𝜈)𝑧 in (28), to get 0 = [𝜈, 𝑟]𝜉(𝜈)𝑧𝜔𝜉(𝜈), that is [𝜈, 𝑟]𝜉(𝜈)ℬ𝑧𝜔𝜉(𝜈) = (0) ∀ 𝜈, 𝜔, 𝑧 ∈ 𝐿, 𝑟 ∈ 𝑅. Interchanging 𝑧 and 𝜔 and then subtracting one from the other, we have [𝜈, 𝑟]𝜉(𝜈)ℬ[𝑧, 𝜔]𝜉(𝜈) = (0) ∀ 𝜈, 𝜔, 𝑧 ∈ 𝐿. In particular, by setting 𝑟 = 𝑧 and 𝜔 = 𝜈, we have [𝜈, 𝑧]𝜉(𝜈)ℬ[𝜈, 𝑧]𝜉(𝜈) = (0), ∀𝜈, 𝑧 ∈ 𝐿. Since ℬ is semiprime, it follows that [𝜈, 𝑧]𝜉(𝜈) = 0 ∀ 𝜈, 𝑧 ∈ 𝐿. (29) By multiplying (29) on the right by 𝜈, we deduce that [𝜈, 𝑧]𝜉(𝜈)𝜈 = 0, ∀𝜈, 𝑧 ∈ 𝐿. Substituting 𝑧 with 𝑧𝜈, we find that [𝜈, 𝑧]𝜈𝜉(𝜈) = 0, ∀𝜈, 𝑧 ∈ 𝐿. Subtracting these two equations yields [𝜈, 𝑧][𝜉(𝜈), 𝜈] = 0, ∀𝜈, 𝑧 ∈ 𝐿. Replacing 𝑧 with 𝜉(𝜈)𝑧 in this result gives [𝜈, 𝜉(𝜈)]𝑧[𝜈, 𝜉(𝜈)] = 0, ∀𝜈, 𝑧 ∈ 𝐿, which can be rewritten as 𝐿[𝜉(𝜈), 𝜈]ℬ𝐿[𝜉(𝜈), 𝜈] = (0), ∀𝜈 ∈ 𝐿. The semiprimeness of ℬ implies that 𝐿[𝜉(𝜈), 𝜈] = (0), ∀𝜈 ∈ 𝐿. From our assumption, we have [𝜈, Γ(𝜈)] = [𝜉(𝜈), 𝜈], and thus 𝐿[𝜈, Γ(𝜈)] = 𝐿[𝜉(𝜈), 𝜈] = 0, ∀𝜈 ∈ 𝐿. The case where [𝜉(𝜈), 𝜔] = −[𝜈, Γ(𝜔)] ∀ 𝜈,𝜔 ∈ 𝐿 follows similarly. Corollary 2.9 Let ℬ be a semiprime ring, and let 𝛤 be a 𝑚𝑢𝑙𝑡. (𝐺) − 𝐷 associated with a map 𝜉. If [𝜉(𝜈), 𝜔] = [𝜈, 𝛤(𝜔)] or [𝜉(𝜈), 𝜔] = −[𝜈, 𝛤(𝜔)], ∀𝜈, 𝜔 ∈ ℬ, then 𝛤 and 𝜉 are commuting maps on ℬ. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2144 https://internationalpubls.com 3- Example The following example demonstrates that the semiprimeness condition in the preceding theorems is essential and cannot be omitted. Example 3.1 Let ℬ = {( 0 𝑎 𝑏 0 0 𝑐 0 0 0 ) |𝑎, 𝑏, 𝑐 ∈ ℤ (𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠)}. For any 0 ≠ 𝑏 ∈ ℤ, ( 0 0 𝑏 0 0 0 0 0 0 )ℬ( 0 0 𝑏 0 0 0 0 0 0 ) = (0), then ℬ is not a semiprime ring. Define 𝛤: ℬ → ℬ and 𝜉: ℬ → ℬ is given by: Γ(( 0 𝑎 𝑏 0 0 𝑐 0 0 0 )) = ( 0 0 𝑏 0 0 0 0 0 0 ), and 𝜉(( 0 𝑎 𝑏 0 0 𝑐 0 0 0 )) = ( 0 𝑎2 0 0 0 𝑐 0 0 0 ). It is easy to verify that Γ is a 𝑀𝑢𝑙𝑡. (𝐺) − 𝐷 associated with the map 𝜉. Furthermore, it can be directly confirmed that: Γ(𝜈)𝜔 = 𝜔Γ(𝜈) = 0, ∀𝜈, 𝜔 ∈ ℬ. Consequently, [Γ(𝜈), 𝜔] = ±[𝜈, Γ(𝜔)], is hold for all 𝜈,𝜔 ∈ ℬ. 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