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Let $R$ be a prime ring with center $Z$ and $I$ be a nonzero ideal of $R$. In this manuscript, we investigate the action of skew derivation $(\delta,\varphi)$ of $R$ which acts as a homomorphism or an anti-homomorphism on $I$. Moreover, we provide an example for semiprime case.

Let R be a prime ring with center Z(R) and I a nonzero right ideal of R. Suppose that R admits a generalized reverse derivation (F, d) such that d(Z(R)) ≠ 0. In the present paper, we shall prove that if one of the following conditions holds: (i) F (xy) ± xy ∈ Z(R) (ii) F ([x, y]) ± [F (x), y] ∈ Z(R) (iii) F ([x, y]) ± [F (x), F (y)] ∈ Z(R) (iv) F (x ο y) ± F (x) ο F (y) ∈ Z(R) (v) [F (x), y] ± [x, F (y)] ∈ Z(R) (vi) F (x) ο y ± x ο F (y) ∈ Z(R) for all x, y ∈ I, then R is commutative.

Let R be a semiprime ring with characteristic different from two and L a noncentral square-closed Lie ideal of R. Suppose that R admits a multiplicative generalized derivation (F,d) satisfying d(L) ⊆ L. In the present paper, we shall prove that d is commuting on L if one of the following conditions holds: (i) F([x,y]) = ∓ [x,d(y)]; (ii) F(x ∘ y) = ∓ (x ∘ d(y)); (iii) F([x,y]) = ∓ (F(y)x); (iv) F([x,y]) = ∓ (F(x)y); (v) F(x ∘ y) = ∓ (F(y)x); (vi) F(x ∘ y) = ∓ (F(x)y) for all x,y ∈ L.

Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $m, n$ fixed positive integers. If $R$ admits a generalized derivation $F$ associated with a nonzero derivation $d$ such that $(F([x,y])^{m}=[x,y]_{n}$ for all $x,y\in I$, then $R$ is commutative. Moreover we also examine the case when $R$ is a semiprime ring.

Let R be a prime ring, I a nonzero ideal of R, and a ∈ R. Suppose that σ is a nontrivial automorphism of R such that a{(σ(x ∘ y))n − (x ∘ y)m} = 0 or a{(σ([x,y]))n − ([x,y])m} = 0 for all x,y ∈ 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.

Bu makalenin amacı yarıasal bir gamma halkasının kesirli gamma halkalarını tanımlamak ve kesirli gamma halkasının bazı özelliklerini araştırmaktır., The aim of this paper is to define the gamma rings of quotients of a semiprime gamma ring and investigate some properties of gamma ring of quotients.

In this paper, using the system of coupled equations involving an auxiliary function, we introduce some new efficient higher order iterative methods based on modified homotopy perturbation method. We study the convergence analysis and also present various numerical examples to demonstrate the validity and efficiency of our methods.

Let R be a 6-torsion free semiprime *-ring, \tau an endomorphism of R, \sigam an epimorphism of R and f : R ! R an additive mapping. In this paper we proved the following result: f is a generalized Jordan ( \sigma,\tau) *-iderivation if and only if f is a generalized Jordan triple ( \sigma,\tau) *-derivation.

Let M be a prime Γ-ring with center Z(M), I a nonzero ideal of M and F be a generalized derivation with associated nonzero derivation d. In the present paper, our purpose is to produce commutativity results for prime Γ-rings M admitting a generalized derivation F satisfying any one of the properties: (i) F(xαy)∓xαy ∈ Z(M), (ii) F(xαy)∓yαx ∈ Z(M), (iii) F(x)αF(y)∓xαy ∈ Z(M), (iv) F(x)αF(y)∓yαx ∈ Z(M), (v) F([x, y]α)=[F(x), y]α, (vi) [F(x), y]α∓[x, F(y)]α=0, (vii) F([x, y]α)∓[d(x), d(y)]α=0, for all x,y ∈ I and α ∈ Γ. Also, some examples are given to show that the primeness of the various results is not superfluous.

The distance, $d(u,v)$, between two vertices $u$ and $v$ of a connected graph $G$ is the length of a $u-v$ geodesic in $G$. A large number of graph-distance-based topological indices in various families of graphs and networks have been computed. In this paper, we consider circulant networks and compute three distance-based topological indices, namely the Wiener index, hyper-Wiener index and Schultz molecular topological index on these networks.

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d[x, y])m = [x, y]n for all x, y ∈ I, then R is commutative. (ii) If Char R ≠ 2 and [d(x), d(y)]m = [x, y]n for all x, y ∈ I, then R is commutative. Moreover, we also examine the case when R is a semiprime ring.

Let R be a prime ring, I a nonzero ideal of R and n a fixed positive integer. If R admits a generalized derivation F associated with a derivation d such that c for all x, . Then either R is commutative or n = 1, d = 0 and F is the identity map on R. Moreover in case R is a semiprime ring and for all x, , then either R is commutative or n = 1, , R contains a non-zero central ideal and for all .

In this paper, we present some results concerning orthogonal generalized ()-derivations in semiprime near-rings. These results are a generalization of result of Bresar and Vukman, which are related to a theorem of Posner for the product of two derivations in prime rings.

The concept of $\Gamma$-ring is a generalization of ring. Twoimportant classes of $\Gamma$-rings are prime and semi-prime$\Gamma$-rings. In this paper, we consider the concept of derivations on prime and semi-prime $\Gamma$-rings and we study some of their properties.

Let R be a prime ring, let L a noncentral Lie ideal, and let f be a nonzero semiderivation associated with an automorphism σ such that f(u)ⁿ = 0 for all u ∈ L; where n is a fixed positive integer. If either Char R > n + 1 or Char R = 0; then R satisfies s₄; the standard identity in four variables. Нехай R — просте кільцє, L — нецентральний ідеал Лі та f — ненульова напівпохідна, асоційована з автоморфiзмом a таким, що f(u)ⁿ=0 для всіх uЄL, де n — фіксоване натуральне число. Якщо Char R>n+1 або Char R=0, то R задовольняє стандартну тотожність s₄ у чотирьох змінних.

Let R be a semiprime ring with center Z(R) and µ be a surjective homomorphism. In this paper, we prove that T is a µ-centralizer if one of the following holds: (i) T(x)µ(y) = µ(x)T(y) for all x;y 2 R, where T is a mapping. (ii) [T(x);µ(x)] = 0 for all x 2 R, where T is a left µ-centralizer. (iii) 2T(xyx) = T(x)µ(y)µ(x) + µ(x)µ(y)T(x) for all x;y 2 R, where T is an additive mapping, µ(Z(R)) = Z(R) and R is 2-torsion free. 1

WOS: 000338514500017, Let (r, *) be a 2-torsion free *-prime ring with involution * and center Z (R), U a nonzero square closed *-Lie ideal of R. An additive mapping F W R! R is called a generalized derivation if there exits a derivation d WR R such that F. xy/DF. x/yCxd. y/. In the present paper, we prove that U * Z. R/if any one of following conditions holds: 1) O F. u/; u _ D 0; 2) O d. u/; F. v/_ D 0; 3) d. u/oF. v/D 0; 4) O d. u/; F. v/ D * O u; v; 5) d. u/oF (v) D * uov; 6) d. u/F. v/* uv 2 Z. R/; for all u; v 2 U: Furthermore, an example is given to demonstrate that the *-primeness hypothesis is not superfluous.