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Let 풜 be a standard operator algebra containing the identity operator I on an infinite dimensional complex Hilbert space ℋ which is closed under adjoint operation. Suppose that ϕ : 풜 → 풜 is the second nonlinear mixed Lie triple derivation. Then ϕ is an additive ∗ -derivation. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with A ⁢ (R) its set of annihilating-ideals. The extended annihilating-ideal graph of R, denoted by AG ¯ ⁢ (R) , is an undirected graph with vertex set A ⁢ (R) * = A ⁢ (R) ∖ { 0 } and two vertices I 1 and I 2 are adjacent if and only if I 1 m ⁢ I 2 n = 0 with I 1 m ≠ 0 and I 2 n ≠ 0 , for some positive integers m and n. In this paper, we first study some basic properties of AG ¯ ⁢ (R) and then we investigate the relationship between the extended annihilating-ideal graph AG ¯ ⁢ (R) , the annihilator-ideal graph A I ⁢ (R) and the annihilating-ideal graph AG ⁢ (R) of a commutative ring R. [ABSTRACT FROM AUTHOR]

For a finite commutative ring R with identity 1 ≠ 0 , the weakly zero-divisor graph of R denoted as W Γ (R) is a simple undirected graph having vertex set as a set of non-zero zero-divisors of R and two distinct vertices a and b are adjacent if and only if there exist elements r ∈ ann (a) and s ∈ ann (b) satisfying the condition r s = 0 . The zero-divisor graph of a ring is a spanning sub-graph of the weakly zero-divisor graph. This article finds the normalized Laplacian spectra of the weakly zero-divisor graph W Γ (R) . Specifically, the investigation is carried out on the weakly zero-divisor graph W Γ (Z n) for various values of n. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, and  Z (R)  be the set of zero-divisors of R. The weakly zero-divisor graph of R denoted by  W Γ (R)  is an undirected (simple) graph with vertex set  Z (R) * , and two distinct vertices x and y are adjacent, if and only if there exist  r ∈ a n n (x)  and  s ∈ a n n (y) , such that  r s = 0 . Importantly, it is worth noting that  W Γ (R)  contains the zero-divisor graph  Γ (R)  as a subgraph. It is known that graph theory applications play crucial roles in different areas one of which is chemical graph theory that deals with the applications of graph theory to solve molecular problems. Analyzing Zagreb indices in chemical graph theory provides numerical descriptors for molecular structures, aiding in property prediction and drug design. These indices find applications in QSAR modeling and chemical informatics, contributing to efficient compound screening and optimization. They are essential tools for advancing pharmaceutical and material science research. This research article focuses on the basic properties of the weakly zero-divisor graph of the ring  Z p × Z t × Z s , denoted by  W Γ (Z p × Z t × Z s) , where p, t, and s are prime numbers that may not necessarily be distinct and greater than 2. Moreover, this study includes the examination of various indices and coindices such as the first and second Zagreb indices and coindices, as well as the first and second multiplicative Zagreb indices and coindices of  W Γ (Z p × Z t × Z s) . [ABSTRACT FROM AUTHOR]

Let q be a power of prime p. In this article, we investigate the reversible cyclic codes of arbitrary length n over the ring R = Fq +uFq + u²Fq, where u³ = 0 mod q. Further, we find a unique set of generators for cyclic codes over R and classify the reversible cyclic codes with their generators. Moreover, it is shown that the dual of reversible cyclic code over R is reversible. Finally, some examples of reversible cyclic codes are provided to justify the importance of these results. [ABSTRACT FROM AUTHOR]

In this work, we study cyclic codes of length n over a finite commutative non-chain ring ℛ = F q [ u , v ] / 〈 u 2 − γ u , v 2 − ϵ v , u v − v u 〉 where γ , ϵ ∈ F q * and we find new and better quantum error-correcting codes than previously known quantum error correcting codes. Then certain constraints are imposed on the generator polynomials of cyclic codes, so these codes become linear complementary dual codes (in short LCD codes). We then verify that the Gray image of linear complementary dual codes of length n over ℛ is a linear complementary dual code of length 4 n over F q by establishing a Gray map. [ABSTRACT FROM AUTHOR]

We define an R-linear map φ from A to an A-bimodule M is said to be Jordan homoderivation if φ(x² ) = φ(x)x+xφ(x) +φ(x)² for each x ∈ A. In this article, we proved that every Jordan homo-derivation to be homo-derivation on triangular algebras. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the commutativity of a prime ring ℛ with anti-automorphism ψ and generalized derivation (ℱ , ξ) satisfying certain algebraic identities. Furthermore, we describe the structure of any additive map which is ψ-centralizing on a 2-torsion free prime ring with anti-automorphism ψ. We also provide examples to show that various conditions imposed on the hypotheses of our results are essential. [ABSTRACT FROM AUTHOR]

This paper concentrates on examining the characterization of nonlinear mixed bi-skew Lie triple ∗- derivations within an ∗-algebra denoted by A which contains a nontrivial projection with a unit I. Additionally, we expand this investigation to applications by describing these derivations within prime ∗-algebras, von Neumann algebras, and standard operator algebras. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove the following result which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let ℒ ⁢ (X) {\mathcal{L}(X)} be the algebra of all bounded linear operators of X into itself and let 𝒜 ⁢ (X) ⊂ ℒ ⁢ (X) {\mathcal{A}(X)\subset\mathcal{L}(X)} be a standard operator algebra. Suppose there exist linear mappings ℋ , 𝒢 : 𝒜 ⁢ (𝒳) → ℒ ⁢ (𝒳) {\mathcal{H},\mathcal{G}\colon\mathcal{A(X)}\to\mathcal{L(X)}} satisfying the relations ℋ ⁢ (𝒜 m + n) = ℋ ⁢ (𝒜 m) ⁢ 𝒜 n + 𝒜 m ⁢ 𝒢 ⁢ (𝒜 n) , \displaystyle\mathcal{H}(\mathcal{A}^{m+n})=\mathcal{H}(\mathcal{A}^{m})% \mathcal{A}^{n}+\mathcal{A}^{m}\mathcal{G}(\mathcal{A}^{n}), 𝒢 ⁢ (𝒜 m + n) = 𝒢 ⁢ (𝒜 m) ⁢ 𝒜 n + 𝒜 m ⁢ ℋ ⁢ (𝒜 n) \displaystyle\mathcal{G}(\mathcal{A}^{m+n})=\mathcal{G}(\mathcal{A}^{m})% \mathcal{A}^{n}+\mathcal{A}^{m}\mathcal{H}(\mathcal{A}^{n}) for all 𝒜 ∈ 𝒜 ⁢ (𝒳) {\mathcal{A}\in\mathcal{A(X)}} and some fixed integers m , n ≥ 1 {m,n\geq 1}. Then there exists ℬ ∈ ℒ ⁢ (𝒳) {\mathcal{B}\in\mathcal{L(X)}} , such that ℋ ⁢ (𝒜) = 𝒜 ⁢ ℬ - ℬ ⁢ 𝒜 {\mathcal{H(A)}=\mathcal{AB}-\mathcal{BA}} for all 𝒜 ∈ ℱ ⁢ (𝒳) {\mathcal{A}\in\mathcal{F(X)}} , where ℱ ⁢ (𝒳) {\mathcal{F(X)}} denotes the ideal of all finite rank operators in ℒ ⁢ (X) {\mathcal{L}(X)} , and ℋ ⁢ (𝒜 m) = 𝒜 m ⁢ ℬ - ℬ ⁢ 𝒜 m {\mathcal{H}(\mathcal{A}^{m})=\mathcal{A}^{m}\mathcal{B}-\mathcal{B}\mathcal{A% }^{m}} for all 𝒜 ∈ 𝒜 ⁢ (𝒳) {\mathcal{A}\in\mathcal{A(X)}}. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with char(R)≠2. A biadditive symmetric map Δ: R×R → R is called symmetric (α, β)-biderivation if, for any fixed y ∈ R, the map x ↦ Δ(x,y) is a (α, β)-derivation. A symmetric biadditive map Γ : R×R → R is a symmetric generalized (α, β)-biderivation if for any fixed y ∈ R, the map x ↦ Γ(x,y) is a generalized (α, β)-derivation of R associated with the (α, β)-derivation Δ(.,y). In the present paper, we investigate the commutativity of a ring having a generalized (α, β)-biderivation satisfying certain algebraic conditions. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2, Z(R) its center, L a Lie ideal of R, and m, n, s, t ≥ 1 fixed integers with t ≤ m + n + s. Suppose that α is a non-trivial automorphism of R and let Φ(x, y) = [x, y]t – [x, y]m [α([x, y]),[x, y]]n [x, y]s. Thus, (a) if Φ(u, v) = 0 for any u, v ∈ L, then L ⊆ Z(R); (b) if Φ(u, v) ∈ Z(R) for any u, v ∈ L, then either L ⊆ Z(R) or R satisfies s4, the standard identity of degree 4. We also extend the results to semiprime rings. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with the extended centroid C and symmetric Martindale quotient ring Qs (R). In this paper we prove the following result. Let F : R → R be a generalized derivation associated with a non-zero derivation d on R and let h be an additive map of R such that F(x)x = xh(x) for all x ∈ R. Then either R is commutative or F(x) = xp and h(x) = px where p ∈ Qs (R). [ABSTRACT FROM AUTHOR]

In this article, we study multiplicative (generalized)-skew derivation G and multiplicative left centralizer H satisfying certain conditions in semiprime rings. Moreover, some examples are given to demonstrate that the semiprimeness imposed on the hypotheses of various theorems is essential. [ABSTRACT FROM AUTHOR]

Let M be a prime Γ-ring, I a nonzero ideal, θ an automorphism and d a θ-derivation of M. In this article we have proved the following result: (1) If d([x, y]α) = ±([x, y]α) or d((x o y)α) = ±((x o y)α) for all x, y ε I;α ε Γ, then M is commutative. (2) Under the hypothesis dθ = θd and Char M, 2, if (d(x) o d(y))α = 0 or [d(x),d(y)]α = 0 for all x, y 2 I;α α Γ, then M is commutative. (3) If d acts as a homomorphism or an antihomomorphism on I, then d = 0 or M is commutative. Moreover, an example is given to demonstrate that the primeness imposed on the hypothesis of the various results is essential. [ABSTRACT FROM AUTHOR]

Let X be a real or complex Banach space, let L ( X ) be the algebra of all bounded linear operators of X and let A ( X ) ⊆ L ( X ) be a standard operator algebra. Suppose there exists a linear mapping T : A ( X ) → L ( X ) satisfying the relation T ( A n ) = T ( A ) A n −1 − AT ( A n −2 ) A − A n −1 T ( A ) for all A ∈ A ( X ) , where n > 2 is some fixed integer. Then T is of the form: (i) T ( A ) = 0 for all A ∈ F ( X ) and (ii) T ( A ) = BA , for all A ∈ A ( X ) and some B ∈ L ( X ) . [ABSTRACT FROM AUTHOR]

Let M be a prime Γ-ring, I a nonzero ideal, θ an automorphism and dd a θ-derivation of MM. In this article we have proved the following result: (1) If d([x,y]α)=±([x,y]α) or d((xoy)α)=±((xoy)α) for x,y∈I;α∈Γ, then M is commutative. (2) Under the hypothesis dθ=θd and CharM ≠ 2, if (d(x) o d(y))α=0 or [d(x),d(y)]α=0 for all x,y∈I;α∈Γ, then M is commutative. (3) If dd acts as a homomorphism or an anti-homomorphism on I, then d=0 or M is commutative. Moreover, an example is given to demonstrate that the primeness imposed on the hypothesis of the various results is essential. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2 and m a fixed positive integer. If R admits a generalized derivation associated with a nonzero deviation d such that [F(x), d(y)]m = [x, y] for all x, y in some appropriate subset of R, then R is commutative. Moreover, we also examine the case R is a semiprime ring. Finally, we apply the above result to Banach algebras, and we obtain a non-commutative version of the Singer--Werner theorem. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic not 2, U a nonzero square closed Lie ideal of R and F;G;H be the generalized derivations with associated derivations d,δh of R respectively. In the present paper, we study the situations if one the follwoing holds (1) F(u)G(v)±H(uv)ϵ Z(R), (2) F(u)F (v)±H(vu) ϵ Z(R), for all u,v ϵ U, then U ⊆Z(R). [ABSTRACT FROM AUTHOR]

Let R be a prime ring and set [x,y]1 = [x,y] = D xy - yx for all x,y ∊ R and inductively [x,y]k = [[x,y]k-1,y] for k > 1. We apply the theory of generalized polynomial identities with automorphism and skew derivations to obtain the following result: Let R be a prime ring and I a nonzero ideal of R. Suppose that (δ,φ) is a skew derivation of R such that δ([x,y])= [x,] = n for all x;y ∊ I , then R is commutative. [ABSTRACT FROM AUTHOR]

Let R be a *-prime ring with characteristic different from two and U ≠ 0 be a square closed *-Lie ideal of R. An additive mapping F : R → R is called an generalized derivation if there exits a derivation d : R → R such that F(xy) = F(x)y+xd(y). In the present paper, it is shown that U ⊆ Z if R is a *-prime ring which admits a generalized derivation satisfying several conditions that are associated with a derivation commuting with *. [ABSTRACT FROM AUTHOR]

Let A be a non-commutative prime ring with involution ∗ , of characteristic ≠ 2 (and 3) , with Z as the center of A and Π a mapping Π : A → A such that [ Π (x) , x ] ∈ Z for all (skew) symmetric elements x ∈ A. If Π is a non-zero CE-Jordan derivation of A , then A satisfies s 4 , the standard polynomial of degree 4. If Π is a non-zero CE-Jordan ∗-derivation of A , then A satisfies s 4 or Π (y) = λ (y − y *) for all y ∈ A , and some λ ∈ C , the extended centroid of A. Furthermore, we give an example to demonstrate the importance of the restrictions put on the assumptions of our results. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2 with extended centroid C. Let F be a generalized derivation of R, L a non-central Lie ideal of R, f(x1, ..., xn) a polynomial over C and f(R)={f(r1, ..., rn): ri ∈ R}. We study the following cases: (1) [F(u), F(v)]k=0 for all u, v ∈ L, where k ≥ 1 is a fixed integer; (2) [F(u), F(v)] = 0 for all u, v ∈ f(R); (3) F(u) ◦ F(v)=0 for all u, v ∈ f(R); (4) F(u) ◦ F(v)=u ◦ v for all u, v ∈ f(R). We obtain a description of the structure of R and information on the form of F. [ABSTRACT FROM AUTHOR]

Introduction: Tibia is one of the long bone of the body. Due to its superficial location tibial fractures are common. Due to increasing population and changing human habits the number of accidents and high energy trauma is rising. Management of open fracture tibia is a complex problem and is a challenge for both Orthopaedic and plastic surgeons and requires the widest experience, the greatest wisdom, and the nicest of the clinical judgment in order to choose the most appropriate treatment for particular pattern of injury. This study will give us the latest and updated knowledge about the frequency of tibial diaphysial fractures after motorcycle accident. Our results may vary with other study because of lack of knowledge of traffic rules, excessive use of motorbike, rash driving. The results of this study will be shared with local orthopedic surgeons to devise future research and policy recommendations. Objective: To determine the frequency of tibial diaphysial fractures among patients presenting with motorcycle accidents. Study design: Descriptive cross sectional. Material and methods: In this study a total of 222 patients were observed. Detail patient history and clinical examination was done. Laboratory investigation and X-rays was confirmed the diagnosis of tibial diaphyseal fractures. All the above mentioned information including name, age, gender, socio-economic status, occupation, education level was recorded in a proforma. Strictly exclusion criteria had followed to control confounders and bias in the study results. Results: In this study mean age was 30 years with SD ± 12.33. 93% patients were male while 7% patients were female. More over 34% patients had tibial diaphyseal fracture and 66% patients did not had tibial diaphyseal fracture. Conclusion: Our study concludes that the frequency of tibial diaphysial fractures was 34% among patients presenting with motorcycle accidents. [ABSTRACT FROM AUTHOR]

Let ℧ be a prime ring of char(℧) ≠ 2 with its center Z. This article introduces new classes of endomorphisms and investigates how they relate to antiautomorphisms of prime rings and the commutativity of prime rings. Additionally, we fully describe and classify some of these endomorphisms. We also give examples to prove the necessity of the numerous restrictions included in the hypotheses of our results. [ABSTRACT FROM AUTHOR]

This article examines the commutativity of rings with antiautomorphisms, specifically when they are equipped with derivations that satisfy certain algebraic identities. Moreover, we present examples to demonstrate the necessity of the various restrictions imposed in the hypotheses of our theorems. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2, C its extended centroid, d a nonzero derivation of R, f (x1, . . . , xn) a multilinear polynomial over C, ρ a nonzero right ideal of R andm > 1 a fixed integer such that ([d(f(r1, . . . , rn)), f(r1, . . . , rn)])m = [d(f(r1, . . . , rn)), f(r1, . . . , rn)] for all r1, . . . , rn ∈ ρ. Then either [f(x1,..., xn), xn+1]xn+2 is an identity for ρ or d(ρ)ρ = 0. [ABSTRACT FROM AUTHOR]

Let R be a prime ring, I a nonzero ideal of R, and a ε R. Suppose that σ is a nontrivial automorphism of R such that af{(σ(x 0 y))n - (x 0 y)m} = 0 or a{(σ([x; y]))n - ([x; y])m} = 0 for all x,y ε I, where n and m are fixed positive integers. We prove that if char(R) > n + 1 or char(R) = 0, then either a = 0 or R is commutative. [ABSTRACT FROM AUTHOR]

Let R be a ring and α,β be endomorphisms of R. An additive mapping F: R → R is called a generalized (α,β)-derivation on R if there exists an (α,β)-derivation d: R → R such that F(xy)=F(x) α(y) + β(x)d(y) holds for all x, y ∈ R. In the present paper, we discuss the commutativity of a prime ring R admitting a generalized (α,β)-derivation F satisfying any one of the properties: (i) [F(x),x]α,β=0, (ii) F([x,y])=0, (iii) F(x ◦ y)=0, (iv) F([x,y])=[x,y]α,β, (v) F(x ◦ y)=(x ◦ y)α,β, (vi) F(xy)- α(xy) ∈ Z(R), (vii) F(x)F(y)- α(xy) ∈ Z(R) for all x, y in an appropriate subset of R. [ABSTRACT FROM AUTHOR]