EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5528 ISSN 1307-5543 – ejpam.com Published by New York Business Global Characterizations of Nonadditive Mappings in Prime ∗-Rings Involving Bi-Skew Products Moin A. Ansari1, Abbas Hussain Shikeh2, Junaid Nisar3,∗, Shahid Tamboli4 1 Department of Mathematics College of Science, Jazan University Jazan 45142, Kingdom of Saudi Arabia 2 Department of Mathematics, Aligarh Muslim University, Aligarh-202002 India 3 Department of Applied Sciences, Symbiosis Institute of Technology, Symbiosis International (Deemed) University, Lavale, Pune, India 4 Department of Mechanical Engineering, Symbiosis Institute of Technology, Symbiosis International (Deemed) University, Lavale, Pune, India Abstract. The paper investigates nonadditive mappings Ω : ℜ → ℜ on a prime ring ℜ with involution ∗, characterized by satisfying one of the following conditions: (i) [Ω(u),Ω(v)]• = [u, v]• for all u, v ∈ ℜ. (ii) [Ω(u), v]• = [u,Ω(v)]• for all u, v ∈ ℜ. (iii) Ω(u • v) = Ω(u) • v for all u, v ∈ ℜ. Furthermore, the paper characterizes generalized bi-skew Jordan derivations within prime ∗-rings and examines the implications of these results in the context of various operator algebras. 2020 Mathematics Subject Classifications: 16N60, 16W10, 47B47 Key Words and Phrases: Involution, prime ring, strong bi-skew commutativity preserving map, bi-skew commuting map, bi-skew Jordan derivation 1. Introduction In this paper, unless stated otherwise, ℜ represents a prime ring with Z(ℜ) as its center. A ring ℜ is defined as prime if, for any u, v ∈ ℜ, the condition uℜv = {0} implies that either u = 0 or v = 0. We denote the maximal left and right rings of quotients of ℜ by Qml(ℜ) and Qmr(ℜ), respectively. The maximal symmetric ring of quotients of ℜ is denoted by Qms(ℜ). It is well established that ℜ ⊆ Qms(ℜ) ⊆ Qml(ℜ) and that Qms(ℜ) = Qml(ℜ) ∩ Qmr(ℜ). Both Qms(ℜ) and Qml(ℜ) are also recognized as prime rings and share a common center, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5528 Email addresses: maansari@jazanu.edu.sa (M. Ansari), abbasnabi94@gmail.com (A. Shikeh), junaidnisar73@gmail.com (J. Nisar), shahidt@sitpune.edu.in (S. Tamboli) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) Moin A. Ansari et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5528 2 of 14 denoted by C, which is the extended centroid of ℜ. The set C is defined as {λ ∈ Qms(ℜ) | λa = aλ for all a ∈ ℜ}. The ring ℜ is prime if and only if C is a field (see [1–3] for more details). An involution ‘∗’ on ℜ is defined as an anti-automorphism of order 1 or 2. An anti- automorphism ∇ of ℜ is classified as being of the first kind with respect to Z(ℜ) if it acts as the identity map on Z(ℜ); otherwise, it is considered of the second kind on Z(ℜ). For a semiprime ring ℜ, if ∇ is an anti-automorphism, then a right ideal I of ℜ is dense if and only if ∇(I) is a dense left ideal of ℜ, and a left ideal J of ℜ is dense if and only if ∇(J ) is a dense right ideal of ℜ. Consequently, with a straightforward modification of the proof in [1, Proposition 2.5.4], it follows that ∇ can be uniquely extended to an anti-automorphism of Qms(ℜ). An anti-automorphism ∇ of ℜ is said to be of the first kind if it acts as the identity on C, and of the second kind otherwise. For elements u, v ∈ ℜ, the following products are defined: the Lie product [u, v] = uv− vu, the skew Lie product [u, v]∗ = uv − vu∗, the skew Jordan product u ⋄ v = uv + vu∗, the bi-skew Lie product [u, v]• = uv∗ − vu∗, and the bi-skew Jordan product u • v = uv∗ + vu∗. These types of products have been the focus of extensive research by various authors (see [4–15]). A map Ω : ℜ → ℜ is termed a strong commutativity preserving map if it satisfies Ω([u, v]) = [u, v] for all u, v ∈ ℜ. In the context of ∗-rings, Ω is referred to as a strong skew commutativity preserving map if Ω([u, v]∗) = [u, v]∗ for all u, v ∈ ℜ, and as a strong bi-skew commutativity preserving map if Ω([u, v]•) = [u, v]• for all u, v ∈ ℜ (see [9, 12, 14, 16]). A map Ω : ℜ → ℜ is called a skew commuting map if it satisfies [Ω(u), v]∗ = [u,Ω(v)]∗ for all u, v ∈ ℜ (see [17]). Similarly, Ω is termed a bi-skew commuting map if [Ω(u), v]• = [u,Ω(v)]• for all u, v ∈ ℜ. A map Ω : ℜ → Qms(ℜ) is described as ∗-linear if it holds that Ω(u∗) = Ω(u)∗ for all u ∈ ℜ. Furthermore, a map Ω : ℜ → ℜ is known as a derivation if it satisfies Ω(uv) = Ω(u)v + uΩ(v) for all u, v ∈ ℜ. In ∗-rings, a map Ω : ℜ → ℜ is called a skew Lie derivation if Ω([u, v]∗) = [Ω(u), v]∗ + [u,Ω(v)]∗ for all u, v ∈ ℜ (see [7, 18]). Similarly, it is called a skew Jordan derivation if Ω(u ⋄ v) = Ω(u) ⋄ v + u ⋄ Ω(v) for all u, v ∈ ℜ. A map Ω : ℜ → ℜ is defined as a bi-skew Jordan derivation if Ω(u • v) = Ω(u) • v+ u •Ω(v) for all u, v ∈ ℜ (see [19–22]). Additionally, a map Ψ : ℜ → ℜ is termed a generalized bi-skew Jordan derivation if there exists a bi-skew Jordan derivation Ω : ℜ → ℜ such that Ψ([u, v]•) = [Ψ(u), v]• + [u,Ω(v)]• for all u, v ∈ ℜ (see [22]). A derivation Ω : ℜ → ℜ is called an additive ∗-derivation if Ω is both additive and ∗-linear. A map Ω : ℜ → ℜ is called a left (resp. right) centralizer if Ω(uv) = Ω(u)v (resp. Ω(uv) = uΩ(v)) for all u, v ∈ ℜ (see [23]). Similarly, Ω is termed a left (resp. right) bi-skew Jordan centralizer if Ω(u • v) = Ω(u) • v (resp. Ω(u • v) = u • Ω(v)) for all u, v ∈ ℜ. Brešar and Miers [24, Theorem 5] proved that if ℜ is a semiprime ring and Ω : ℜ → ℜ is an additive strong commutativity preserving map, then Ω(u) = λu+µ(u) for all u ∈ ℜ, where λ ∈ C and µ : ℜ → C is an additive map. Recently, several authors have studied strong commutativity preserving maps (see [8, 10, 11, 15]). Strong skew commutativity preserving maps have received a lot of attention from various algebraists and have been widely studied in the context of rings and algebras (see [4, 6, 9, 12, 14]). Quiet recently, Siddeeque et al. [25, Theorem 2.2] characterized surjective strong skew commutativity Moin A. Ansari et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5528 3 of 14 preserving maps in prime rings without assuming the existence of the unity and a nontrivial symmetric idempotent. They established that if ℜ is a prime ring with an involution ‘∗’ and Ω : ℜ → ℜ is a surjective strong skew commutativity preserving map, then there exists λ ∈ {1,−1} such that Ω(a) = λa for all a ∈ ℜ. Qi and Chen [16, Theorem 2.1] characterized surjective strong bi-skew commutativity preserving maps on prime ∗- algebras. Consequently, they proved the following result: Let A be a prime ∗-algebra, over a field K, with unity I and containing a nontrivial symmetric idempotent. Suppose that ‘∗’ is of the second kind on Z(A) and Ψ(I)Ψ(I)∗ = Ψ(I)∗Ψ(I) = I. If Ψ : A → A is a surjective strong bi-skew commutativity preserving map, then Ψ(u) = αuΨ(I) for all u ∈ A, where α∗ = α ∈ C and α2 = I. Khong and Zhang [17] showed, under certain restrictions, that if ℜ is a unital ∗-ring containing a nontrivial symmetric idempotent and Ω : ℜ → ℜ is a skew commutating map, then there exists λ∗ = λ ∈ Z(ℜ) such that Ω(a) = λa for all a ∈ ℜ. Recently, Siddeeque et al. [25, Theorem 2.3] characterized skew commutating maps in prime rings without assuming the existence of the unity and a nontrivial symmetric idempotent. They established that if ℜ is a prime ring with an involution ‘∗’ and Ω : ℜ → ℜ is a skew commutating map, then there exists λ∗ = λ ∈ C such that Ω(a) = λa for all a ∈ ℜ. Motivated by the above results, in Section 2 of the present paper, we will characterize surjective strong bi-skew commutativity preserving maps and bi-skew commutating maps in prime rings without assuming the existence of the unity and a nontrivial symmetric idempotent (see Theorems 2.1 and 2.2). As applications, we will characterize such maps in different operator algebras. Skew Lie, skew Jordan and bi-skew Jordan derivations have been explored by various algebraists in the context of algebras and rings (see [7, 18, 21, 22] and their bibliographic content). Very recently, Siddeeque and Shikeh [20] characterized bi-skew Jordan deriva- tions in prime rings and proved that every bi-skew Jordan derivation on a unital prime ∗-ring containing a nontrivial symmetric idempotent is an additive ∗-derivation. In Section 3, we will characterize generalized bi-skew Jordan derivations in prime rings (see Theo- rem 3.2). As applications, we will characterize generalized bi-skew Jordan derivations in different operator algebras. 2. Strong bi-skew commutativity preserving maps and bi-skew commuting maps in prime rings We facilitate our discussion with the following lemma which plays a crucial role in the proof of our main results. Lemma 2.1. Let ℜ be a prime ring with an involution ‘∗’ of order 2 and let a, b ∈ Qml(ℜ) such that bu∗ = ua for all u ∈ ℜ. Then a = b = 0. Proof. If ℜ is noncommutative, then by [20, Lemma 2.1], the result follows. Therefore, let’s assume that ℜ is commutative. Thus, α∗ ̸= α for some α ∈ ℜ. Substituting αu for u in the given relation, we find that bα∗u∗ = αua for all u ∈ ℜ. Also, αbu∗ = αua for all u ∈ ℜ. Hence, (α∗ − α)bu∗ = 0 for all u ∈ ℜ. Therefore, b = 0 and consequently, a = 0. Moin A. Ansari et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5528 4 of 14 Lemma 2.2. Let ℜ be a prime PI-ring with an anti-automorphism ∇. Then ∇ is of first kind if and only if ∇ is of the first kind on Z(ℜ). Proof. By [26, Corollary 1], Qml(ℜ) = ℜC = {u α | u ∈ ℜ and 0 ̸= α ∈ Z(ℜ)}. Therefore if ∇ is of the first kind on Z(ℜ), then ∇ can be uniquely extended to an anti- automorphism of ℜC, denoted by ∇ also, by defining ∇( a α ) = ∇(u) α . Hence ∇ is of the first kind. The converse holds trivially. Lemma 2.3. [20, Lemma 2.2] Let ℜ be a ring with an involution ‘∗’ and let Ω : ℜ×ℜ → G be a map, where G is an additive group. Suppose f, h : ℜ → G are maps such that Ω(u, v) = f(uv∗)+h(vu∗) for all u, v ∈ ℜ. Then Ω(uw, v) = Ω(u, vw∗) for all u,w, v ∈ ℜ. The following result provides a characterization of strong bi-skew commutativity pre- serving maps in prime rings without assuming the existence of the unity and a nontrivial symmetric idempotent, thereby generalizing, improving and extending [16, Theorem 2.1] to prime rings. Theorem 2.1. Let ℜ be a prime ring with an involution ‘∗’ of order 2, and let Ω : ℜ → ℜ be a surjective strong bi-skew commutativity preserving map. Then there exists q ∈ Qms(ℜ) such that qq∗ = 1 and Ω(u) = uq for all u ∈ ℜ provided that either dimCℜC > 4 or both char(ℜ) ̸= 2 and ’∗‘ is of the second kind. Proof. By the given hypothesis , we have [Ω(u),Ω(v)]• = [u, v]• (2.1) for all u, v ∈ ℜ. Firstly we establish some facts about Ω. Fact I. Ω is additive. For every u, v, w ∈ ℜ, we have [Ω(u),Ω(v + w)− Ω(v)− Ω(w)]• = [Ω(u),Ω(v + w)]• − [Ω(u),Ω(v)]• − [Ω(u),Ω(w)]• = [u, v + w]• − [u, v]• − [u,w]• = 0. Therefore by the surjectiveness of Ω, we find that [u,Ω(v + w)− Ω(v)− Ω(w)]• = 0 for all u, v, w ∈ R. Now in view of Lemma 2.1, it follows that Ω(u+ v) = Ω(u) + Ω(v) for all u, v ∈ ℜ, that is, Ω is additive. Fact II. Ω is injective. Let a ∈ ℜ be such that Ω(a) = 0. then [a, v]• = [Ω(a),Ω(v)]• = 0 for all v ∈ ℜ. Hence by Lemma 2.1, a = 0. Thus Ω is injective. Therefore from (2.1), we have Ω(u)v∗ +Ω−1(v)u∗ − uΩ−1(v)∗ − vΩ(u)∗ = 0, (2.2) Moin A. Ansari et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5528 5 of 14 for all u, v ∈ ℜ. Next, we advance by examining the following two scenarios: Case I. dimCℜC > 4. In view of [27, Theorem 3.5], it follows that there exists q ∈ Qml(ℜ) such that Ω(u) = uq for all u ∈ ℜ and Ω−1(v)∗ = qv∗ for all v ∈ ℜ. Note that qv∗ = Ω−1(v)∗ ∈ ℜ for all v ∈ ℜ. Hence qℜ ⊆ ℜ. Consequently, q ∈ Qms(ℜ). Therefore Ω−1(u) = uq∗ for all u ∈ ℜ and hence Ω(u)q∗ = u for all u ∈ ℜ. Using Ω(u) = uq in the last relation, we get uqq∗ = u for all u ∈ ℜ. This entails that qq∗ = 1. Case I. dimCℜC ≤ 4. In this case ℜ is a PI-ring and Qml(ℜ) = Qms(ℜ) = ℜC. According to the given hypothesis char(ℜ) ̸= 2 and ’∗‘ is of the second kind. Hence by Lemma 2.2, there is ζ ∈ Z(ℜ) such that ζ∗ ̸= ζ. Let α = ζ∗ − ζ and β = α2. Then α∗ = −α and β∗ = β. Setting v = α in (2.2), we get α(Ω(u) + Ω(u)∗) = Ω−1(α)u∗ − uΩ−1(α)∗, (2.3) for all u ∈ ℜ. Also taking v = β in (2.2), we get β(Ω(u)− Ω(u)∗) = uΩ−1(β)∗ − Ω−1(β)u∗ (2.4) for all u ∈ ℜ. Multiplying both sides of (2.3) by β and (2.4) by α, we get αβ(Ω(u) + Ω(u)∗) = βΩ−1(α)u∗ − βuΩ−1(α)∗, (2.5) and αβ(Ω(u)− Ω(u)∗) = αuΩ−1(β)∗ − αΩ−1(β)u∗ (2.6) for all u ∈ ℜ, respectively. Adding (2.5) and (2.6), we find that 2αβΩ(u) = αuΩ−1(β)∗ − αΩ−1(β)u∗ + βΩ−1(α)u∗ − βuΩ−1(α)∗, (2.7) for all u ∈ ℜ. Now 2αβ is invertible in C. Hence, we have Ω(u) = uq + q1u ∗, (2.8) for all u ∈ ℜ, where q = (2αβ)−1(αΩ−1(β)∗−βΩ−1(α)∗) ∈ ℜC and q1 = (2αβ)−1(βΩ−1(α)− αΩ−1(β)) ∈ ℜC. Using this in (2.1), we get (uq + q1u ∗)(q∗v∗ + vq∗1)− (vq + q1v ∗)(q∗u∗ + uq∗1) = uv∗ − vu∗, (2.9) for all u, v ∈ ℜ. Replacing v by αv in (2.9), we get (uq + q1u ∗)(−q∗v∗ + vq∗1)− (vq − q1v ∗)(q∗u∗ + uq∗1) = −uv∗ − vu∗, (2.10) for all u, v ∈ ℜ. Subtracting (2.10) and (2.9), we find that (uq + q1u ∗)q∗v∗ − q1v ∗(q∗u∗ + uq∗1) = uv∗, (2.11) Moin A. Ansari et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5528 6 of 14 for all u, v ∈ ℜ.Alter u by αu in (2.11), we have (uq − q1u ∗)q∗v∗ − q1v ∗(−q∗u∗ + uq∗1) = uv∗, (2.12) for all u, v ∈ ℜ. Subtracting (2.12) from (2.11), we obtain q1u ∗q∗v∗ = q1v ∗q∗u∗, (2.13) for all u, v ∈ ℜ. Setting u = α in (2.13), we get q1[q ∗, v] = 0 for all v ∈ ℜ. Therefore, q1 = 0 or q ∈ C. If q1 = 0, then from (2.11), we find that u(qq∗ − 1)v = 0 for all u, v ∈ ℜ. Consequently, qq∗ = 1. Hence we assume that q ∈ C. Adding (2.11) and (2.12), we obtain qq∗uv − q1vuq ∗ 1 = uv, (2.14) for all u, v ∈ ℜ. Putting u = v = α in (2.14), we find that qq∗ − q1q ∗ 1 = 1. (2.15) Hence from (2.14), we have q1q ∗ 1uv = q1vuq ∗ 1 for all u, v ∈ ℜ. Taking v = α in the last relation, we have q1[q ∗ 1, u] = 0 for all u ∈ ℜ. Therefore q1 ∈ C and hence from (2.14), we find that qq∗uv − q1q ∗ 1vu = uv for all u, v ∈ ℜ. Using (2.15), we get q1q ∗ 1(uv + vu) = 0 for all u, v ∈ ℜ. Putting u = v = α in the previous relation, we get q1q ∗ 1 = 0. Since C is a field, we conclude that q1 = 0. From (2.8), we infer that Ω(u) = uq for all u ∈ ℜ. The following example shows that Theorem 2.1 does not hold if dimCℜC = 4 and ‘∗’ is of the first kind. Example 2.1. Consider the ring M2(K) of all 2 × 2 matrices over any field K with involution ∗ as the usual transpose map. Let λ ∈ K be fixed such that λ /∈ {−1, 0, 1}. Define the map Ω : M2(K) → M2(K) by Ω ([ a b c d ]) = [ λa b λ c λ λd ] . Then it can be easily seen that Ω([u, v]•) = [u, v]• holds for all u, v ∈ M2(K). However, Ω is not of the form as described in Theorem 2.1. The following result gives a characterization of bi-skew commuting maps in prime rings without assuming the existence of the unity and a nontrivial symmetric idempotent. Theorem 2.2. Let ℜ be a prime ring with an involution ‘∗’ of order 2 and let Ω : ℜ → ℜ be a bi-skew commuting map. Then there exists q∗ = q ∈ Qms(ℜ) such that Ω(u) = uq for all u ∈ ℜ provided that either dimCℜC > 4 or both char(ℜ) ̸= 2 and ’∗‘ is of the second kind. Proof. According to the given hypothesis [Ω(u), v]• = [u,Ω(v)]• (2.16) for all u, v ∈ ℜ. First we show Ω is additive. For every u, v, w ∈ ℜ, we have [Ω(u+ w)− Ω(u)− Ω(w), v]• = [Ω(u+ w), v]• − [Ω(u), v]• − [Ω(w), v]• Moin A. Ansari et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5528 7 of 14 = [u+ w,Ω(v)]• − [u,Ω(v)]• − [w,Ω(v)]• = 0 Applying Lemma 2.1, we infer that Ω(u+ v) = Ω(u) + Ω(v) for all a, b ∈ ℜ, that is, Ω is additive. Now (2.16), can be rewritten as uΩ(v)∗ + vΩ(u)∗ − Ω(u)v∗ − Ω(v)u∗ = 0 (2.17) for all u, v ∈ ℜ. Now we proceed by considering the following two cases:- Case 1: dim(ℜC) > 4. According to [27, Theorem 3.5] there exists q ∈ Qml(ℜ) such that Ω(u) = uq for all u ∈ ℜ (2.18) and Ω(v)∗ = qv∗ for all v ∈ ℜ. (2.19) Since qv∗ = Ω(v)∗ ∈ ℜ for all v ∈ ℜ. Therefore qℜ ⊆ ℜ and hence q ∈ Qmr(ℜ). Consequently, q ∈ Qms(ℜ). Thus from (2.19), we find that Ω(u) = uq∗ for all u ∈ ℜ. (2.20) From (2.18) and (2.20) , we have uq = uq∗ for all u ∈ ℜ. Hence q∗ = q. Case II: dim(ℜ) ≤ 4. In this case by [26, Theorem 2], Z(A) ̸= {0}. Also, by the given hypothesis char(ℜ) ̸= 2 and ζ∗ ̸= ζ for some ζ ∈ Z(A). Let α = ζ∗ − ζ and β = α2. Then α∗ = −α and β∗ = β. Setting v = α in (2.17), we get α(Ω(u) + Ω(u)∗) = Ω(α)u∗ − uΩ(α)∗, (2.21) for all u ∈ ℜ. Also taking v = β in (2.17), we get β(Ω(u)− Ω(u)∗) = uΩ(β)∗ − Ω(β)u∗ (2.22) for all u ∈ ℜ. Multiplying both sides of (2.21) by β and (2.22) by α, we get αβ(Ω(u) + Ω(u)∗) = βΩ(α)u∗ − βuΩ(α)∗, (2.23) and αβ(Ω(u)− Ω(u)∗) = αuΩ(β)∗ − αΩ(β)u∗ (2.24) for all u ∈ ℜ, respectively. Adding (2.23) and (2.24), we find that Ω(u) = uq + q1u ∗, (2.25) for all u ∈ ℜ, where q = (2αβ)−1(αΩ(β)∗ − βΩ(α)∗) ∈ ℜC and q1 = (2αβ)−1(βΩ(α) − αΩ(β)) ∈ ℜC. Using this in (2.17), we have u(vq∗1 + q∗v∗) + v(uq∗1 + q∗u∗)− (uq + q1u ∗)v∗ − (vq + q1v ∗)u∗ = 0 (2.26) Moin A. Ansari et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5528 8 of 14 for all u, v ∈ ℜ.Alter v by αv in (2.26), we get u(vq∗1 − q∗v∗) + v(uq∗1 + q∗u∗) + (uq + q1u ∗)v∗ − (vq − q1v ∗)u∗ = 0 (2.27) for all u, v ∈ ℜ. Adding (2.26) and (2.27), we find that uvq∗1 + v(uq∗1 + q∗u∗)− vqu∗ = 0 (2.28) for all u, v ∈ ℜ.Alter u by αu in (2.28), we get uvq∗1 + v(uq∗1 − q∗u∗) + vqu∗ = 0 (2.29) for all u, v ∈ ℜ. Adding (2.28) and (2.29), we get (uv + vu)q∗1 = 0 for all u, v ∈ ℜ. Consequently, q1 = 0 and hence from (2.28), we see that vq∗u = vqu for all u, v ∈ ℜ. Thus q∗ = q. Hence Ω(u) = qu for all u ∈ ℜ, where q∗ = q ∈ ℜC. The following example shows that Theorem 2.2 does not hold if dimCℜC = 4 and ‘∗’ is of the first kind. Example 2.2. Let M2(K) be the ring of all square matrices of order 2 over any field K with involution ‘∗’ as the usual transpose. Define the map Ω : M2(K) → M2(K) by Ω ([ ζ1 ζ2 ζ3 ζ4 ]) = [ ζ1 − ζ4 ζ2 − ζ3 ζ2 − ζ3 ζ1 − ζ4 ] . Then it can be easily seen that [Ω(a), b]• = [a,Ω(b)]• for all a, b ∈ ℜ. However, Ω is not of the form as described in Theorem 2.2. 3. Generalized bi-skew Jordan derivations in prime rings The following result gives a characterization of additive bi-skew Jordan derivations in prime rings without assuming the existence of a nontrivial symmetric idempotent. Theorem 3.1. Let ℜ be a noncommutative prime ring with an involution ‘∗’ and let Ω : ℜ → Qms(ℜ) be an additive bi-skew Jordan derivation. Suppose that either dimCℜC > 4 or ℜ is unital. Then Ω is a ∗-derivation unless dimCℜC = 4 and char(ℜ) = 2. Proof. Since Ω : ℜ → Qms(ℜ) is an additive bi-skew Jordan derivation, therefore Ω(u • v) = Ω(u) • v + u • Ω(v) for all u, v ∈ ℜ. Now a • b = b • a for any a, b ∈ ℜ. Hence if ℜ is 2-torsion free, then replacing v by u in the previous expression, we have Ω(uu∗) = Ω(u) • u for all u ∈ ℜ. Also if Ω : ℜ → Qms(ℜ) is an additive map satisfying Ω(uu∗) = Ω(u) • u for all u ∈ ℜ, then replacing u by u+ v in the last expression, we find that Ω(u • v) = Ω(u) • v + u • Ω(v) for all u, v ∈ ℜ. Thus if ℜ is a 2-torsion free prime ring with an involution ‘∗’, then an additive map Ω : ℜ → Qms(ℜ) is a bi-skew Jordan derivation if and only if Ω(uu∗) = Ω(u)•u for all u ∈ ℜ. Therefore in view of Theorem 2.1 of [20] and the arguments given in Case I of its proof, we conclude that Ω is a ∗-derivation unless dimCℜC = 4 and char(ℜ) = 2. Proposition 3.1. Let ℜ be a noncommutative prime ring with an involution ‘∗’ and let Ω : ℜ → Qms(ℜ) be a left (right) bi-skew Jordan centralizer. Then there exists λ∗ = λ ∈ C such that Ω(u) = λu for all u ∈ ℜ. Moin A. Ansari et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5528 9 of 14 Proof. We give the details of the proof only when Ω is a left bi-skew Jordan centralizer. The case when Ω is a right bi-skew Jordan centralizer can be proved by using similar arguments. Suppose Ω(u • v) = Ω(u) • v (3.1) for all u, v ∈ ℜ. Alter v by v + w in (3.1), we have Ω(u • v + u • w) = Ω(u) • v +Ω(u) • w (3.2) for all u, v, w ∈ ℜ. That is, Ω(v • u+ v • w) = Ω(v) • u+Ω(v) • w (3.3) for all u, v, w ∈ ℜ. Also alter u by u+ w in (3.1), we have Ω(u • v + w • v) = Ω(u+ w) • v (3.4) for all u, v, w ∈ ℜ. Comparing (3.3) and (3.4), we find that Ω(u+ w) • v = Ω(v) • u+Ω(v) • w (3.5) for all u, v, w ∈ ℜ. Putting w = 0 in (3.5), we get Ω(u) • v = Ω(v) • u (3.6) for all u, v ∈ ℜ. Therefore from (3.5), we find that (Ω(u+ w)− Ω(u)− Ω(w)) • v = 0 (3.7) for all u, v, w ∈ ℜ. In view of Lemma 2.1, it follows that Ω(u+ w) = Ω(u) + Ω(w) for all u, v ∈ ℜ. Thus Ω is additive. Now (3.1) can be rewritten as Ω(uv∗ + vu∗) = Ω(u)v∗ + vΩ(u)∗ (3.8) for all u, v ∈ ℜ. Consider the map Φ : ℜ2 → Qms(ℜ) given by Φ(u, v) = Ω(uv∗) + Ω(vu∗). In view of Lemma 2.3, it follows that Φ satisfies the following relation Φ(uw, v) = Φ(u, vw∗) for all u, v, w ∈ ℜ. Using (3.8), we obtain (Ω(uw)− Ω(u)w)v∗ + v(Ω(uw)∗ − w∗Ω(u)∗) = 0 (3.9) Moin A. Ansari et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5528 10 of 14 for all u, v, w ∈ ℜ. Applying Lemma 2.1, we deduce that Ω(uw) = Ω(u)w for all u,w ∈ ℜ. By [23, Lemma 2.1] there exists q ∈ Qmr(ℜ) such that Ω(u) = qu for all u ∈ ℜ. Now in view of [1, Proposition 2.1.7], it follows that there exists a nonzero dense right ideal K of ℜ such that Kq ⊆ ℜ. Hence from (3.8), we have qvu∗ = v(qu)∗ for all u ∈ K and v ∈ ℜ. (3.10) For each fixed u this is a GPI. Hence by [1, Theorem 6.4.4] qvu∗ = v(qu)∗ for all v ∈ Qmr(ℜ). Therefore putting v = 1, we find that qu∗ = (qu)∗ for all u ∈ K. Thus from (3.10), we have qvu∗ = vqu∗ for all v ∈ ℜ and u ∈ K. Consequently, q ∈ C. Now from (3.10) it can be easily seen that q∗ = q. This completes the proof. Now we are ready to provide a characterization of generalized bi-skew Jordan deriva- tions in prime rings. Theorem 3.2. Let ℜ be a unital prime ∗-ring containing a nontrivial symmetric idem- potent. Suppose that Φ : ℜ → Qms(ℜ) is a generalized bi-skew Jordan derivation with Ω : ℜ → Qms(ℜ) as associated bi-skew Jordan derivation. Then Ω is an additive ∗- derivation and there exists λ∗ = λ ∈ C such that Φ(u) = λu + Ω(u) unless dimCℜC = 4 and char(ℜ) = 2. Proof. By the given hypothesis Φ(u • v) = Φ(u) • v + u • Ω(v) and Ω(u • v) = Ω(u) • v + u • Ω(v) for all u, v ∈ ℜ. Therefore, Ψ(u • v) = Ψ(u) • v for all u, v ∈ ℜ, where Ψ : ℜ → Qms(ℜ) is a map given by Ψ(u) = (Φ − Ω)(u). In view of Proposition 3.1, it follows that there exists λ∗ = λ ∈ C such that Ψ(u) = λu for all u ∈ ℜ. Consequently, Φ(u) = λu + Ω(u) for all u ∈ ℜ. Finally, by [20], Ω is an additive ∗-derivation. This completes the proof. The following result gives a characterization of generalized additive bi-skew Jordan derivations in prime rings without assuming the existence of a nontrivial symmetric idem- potent. Theorem 3.3. Let ℜ be a noncommutative prime ring with an involution ‘∗’ and let Φ : ℜ → Qms(ℜ) be a generalized bi-skew Jordan derivation with Ω : ℜ → Qms(ℜ) as associated bi-skew Jordan derivation such that both Φ and Ω are additive. Suppose that either dimCℜC > 4 or ℜ is unital. Then Ω is a ∗-derivation and there exists λ∗ = λ ∈ C such that Φ(u) = λu+Ω(u) unless dimCℜC = 4 and char(ℜ) = 2. Moin A. Ansari et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5528 11 of 14 Proof. By the given hypothesis Φ(u • v) = Φ(u) • v + u • Ω(v) and Ω(u • v) = Ω(u) • v + u • Ω(v) for all u, v ∈ ℜ. Therefore, Ψ(u • v) = Ψ(u) • v for all u, v ∈ ℜ, where Ψ : ℜ → Qms(ℜ) is a map given by Ψ(u) = (Φ − Ω)(u). In view of Proposition 3.1, it follows that there exists λ∗ = λ ∈ C such that Ψ(u) = λu for all u ∈ ℜ. Consequently, Φ(u) = λu + Ω(u) for all u ∈ ℜ. Finally, by Theorem 3.1, Ω is an ∗-derivation. This completes the proof. 4. Applications to Some Operator Algebras As applications of the outcomes presented in the preceding sections, we aim to delineate strong biskew commutativity-preserving maps, bi-skew commuting maps, and generalized bi-skew Jordan derivations within standard operator algebras and factor von Neumann algebras. Throughout this section, all vector spaces and algebras are defined over the field C of complex numbers. Suppose H represents a Hilbert space, with B(H) denoting the algebra comprising all bounded linear operators on H, and F(H) representing the ideal consisting of all finite rank operators within B(H). The map T 7→ T ∗, which takes an operator to its Hilbert adjoint operator, is an involution on B(H). Here C, the field of complex numbers, is equipped with the conjugate involution and B(H) forms a ∗-algebra. Therefore B(H) is an algebra of characteristic zero and the map T 7→ T ∗, where T ∗ denotes the Hilbert adjoint operator of T , is an involution of the second kind on Z(B(H)). A subset M of B(H) is said to be closed under adjoint operation if u ∈ M implies that u∗ ∈ M, that is, M∗ ⊆ M. Standard operator algebras:A subalgebra S of B(H) earns the label of a standard operator algebra if it includes the identity operator and encompasses F(H). It’s evident that B(H) itself is a standard operator algebra. Furthermore, every standard operator algebra qualifies as a prime algebra. Moreover, for any standard operator algebra S, its center Z(S) is CI. A self-adjoint standard operator algebra S represents an algebra of characteristic zero, and a mapping T 7→ T ∗, where T ∗ denotes the Hilbert adjoint operator of T , serves as an involution of the second kind on Z(S). Leveraging the results derived in the preceding section, the following corollaries emerge. Corollary 4.1. Let S be a self-adjoint standard operator algebra on a Hilbert space H. Suppose that χ : S → S is a surjective map. Then χ is strong bi-skew commutativity preserving map if and only if there exists λ ∈ Qms(S) with λλ∗ = 1 such that χ(a) = λa for all a ∈ S. Moin A. Ansari et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5528 12 of 14 Corollary 4.2. Let S be a self-adjoint standard operator algebra on a Hilbert space H. Then χ : S → S is a bi-skew commuting map if and only if there exists λ ∈ R such that χ(a) = λa for all a ∈ S. Corollary 4.3. Let S be a self-adjoint standard operator algebra on a Hilbert space H. Suppose that Φ : S → S is a generalized bi-skew Jordan derivation with Ω : ℜ → S as associated bi-skew Jordan derivation. Then Ω : S → S is an additive ∗-derivation and there exists λ ∈ R such that Φ(u) = λu+Ω(u). Factor von Neumann algebras A von Neumann algebra N is a subalgebra of B(H) which satisfies the double commutant property, that is, N ′′ = N where N ′ = {T ∈ B(H) | TF = FT for all F ∈ N} and M′′ = (M′)′. It is clear that a von Neumann algebra is unital. A von Neumann algebra N is an algebra of characteristic zero and a map T 7→ T ∗, where T ∗ denotes the Hilbert adjoint operator of T , is an involution of the second kind on Z(N ). A von Neumann algebra N is called a factor von Neumann algebra if its center is trivial, that is, Z(N ) = CI. Every factor von Neumann algebra is a prime algebra. Corollary 4.4. Let N be a factor von Neumann algebra. Suppose that χ : N → N is a surjective map. 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