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The concept of $\Gamma$-ring is a generalization of ring. Two important 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, $I$ 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 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) *-¡derivation if and only if f is a generalized Jordan triple ( \sigma,\tau) *-derivation.

Let R be a 2-torsion-free prime ring with center Z(R), F a generalized derivation associated with a nonzero derivation d, L a Lie ideal of R. If (d(u)l1F(u)l2d(u)l3F(u)l4…F(u)lk)n=0 for all u ∈ L, where l1, l2, … , lk are fixed non-negative integers not all zero, and n is fixed positive integer, then L ⊆ Z(R). We also examine the case when R is a semiprime ring. Finally, we apply the above result to Banach algebras.

Let R be a 2-torsion free sigma-prime ring with an involution sigma, I a nonzero sigma-ideal of R. In this paper we explore the commutativity of R satisfying any one of the properties: (i)d(x)◦F(y) = 0 for all x, y ∈ I. (ii) [d(x),F(y)] = 0 for all x,y ∈ I. (iii) d(x)◦F(y) = x◦y for all x, y ∈ I. (iv) d(x)F(y) − xy ∈ Z(R) for all x, y ∈ I. We also discuss (alpha,beta)-derivations of sigma-prime rings and prove that if G is an (alpha,beta)-derivation which acts as a homomorphism or as an anti-homomorphism on I, then G = 0 or G = on I.

NHR-200-II is a small integrated pressurized water reactor with 200 MW core thermal power. The core heat is transferred to two independent intermediate circuits via fourteen in-vessel primary heat exchangers (PHE), and the heat in the intermediate circuits is transferred to feedwater by two steam generators (SG) in the two intermediate circuits respectively. A passive residual heat removal (PRHR) branch is connected to each intermediate circuit to remove core decay heat under postulated accidents. During normal operation, PRHR branches are isolated by valves while SG branches in intermediate circuits are open. The valves in PRHR branches will be opened and the isolation valves of SG branches will be closed during decay heat removal scenarios. The decay heat removal capacity of NHR-200-II PRHRS could be seriously deteriorated once the isolation valves for SG branches fail to close, which was confirmed in a scaled integral test loop previously. Current understanding of PRHRS's thermal- hydraulic characteristics with possible isolation failure in SG branches is limited. In this paper, the NHR-200-II PRHRS is modeled with RELAP5 considering the case of success and fail to isolate SG branches. A series of numerical simulations are carried out to study the impact of various parameters, such as the initial temperature, the size of the intermediate circuits' header, and the initial flow direction in the intermediate circuits. Oscillatory flow is found when SG branches fail to be isolated under certain parameters combinations. An improved PRHRS design is purposed to eliminate possible flow oscillations, and the purposed improved design are tested by numerical simulations. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with center Z(R), central closure RC, right Martindale quotient ring Qr, F a left generalized derivation and I a nonzero right ideal of R which is semiprime as a ring. We proved in this article that if one of the following conditions holds: (i) F(x o y) ± x o y = 0 (ii) F(xy) ± xy ∈ Z(R) (iii) F(x)F(y) ± xy ∈ Z(R) (iv) F([x,y]) = ±[F(x), y] (v) F(x o y) = ±(F(x) o y) for all x, y ∈ I, then R is commutative or there exists q ∈Qr(RC) such that F(x) = qx for all x ∈ R. [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 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 o y) = ±(x o d(y)) (iii)F([x, y]) = ±(F(y)x) (iv)F([x, y]) = ±(F(x)y) (v)F(x o y) = ±(F(y)x) (vi) F(x o y) = ±(F(x)y) for all x, y 2 L. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with center Z(R), and d a derivation of R. Suppose that (d[x, y]k)n - m[x, y]k ∈ Z(R) for all x, y ∈ R, where m≠ n, k ≥ 1 are fixed integers. Then d = 0 or R satisfies s4, the standard identity in four variables. In the case (d[x, y]kn - m[x, y]k = 0 for all x, y ∈ R, then d = 0 or R is commutative. [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 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 2-torsion free *-prime ring with involution * and center Z(R), U a nonzero square closed *-Lie ideal of R. An additive mapping F : R→R is called a generalized derivation if there exits a derivation d : R→R such that F(xy)=F(x)y+xd(y). In the present paper, we prove that U ⊆ Z(R) if any one of following conditions holds: 1) [F (u), u] = 0, 2) [d(u), F(v) = 0, 3) d(u)oF(v) = 0; 4) [d(u), F(v)] = ±[u,v], 5) d(u)oF(v) = ±uov, 6) d(u)F(v)±uv ϵ Z(R), for all u, v ϵ U. Furthermore, an example is given to demonstrate that the *-primeness hypothesis is not superfluous. [ABSTRACT FROM AUTHOR]

Let N be a prime near-ring and σ, τ be automorphisms of N. Suppose that F is a nonzero left generalized (σ, σ) -derivation of N associated with a nonzero (σ, σ)-derivation d. In the present paper, our objective is to prove that N is a commutative ring if one of the following conditions holds: (i) F([x, y]) = 0, (ii) F(x o y) = 0, (iii) F([x,y]) = σ([x, y]), (iv) F([x,y]) = σ(-xy + yx) for all x,y ∈ N. Further, suppose that F is a left generalized (σ, τ)-derivation of N associated with a nonzero (σ, τ)-derivation d. If F acts as a homomorphism or an anti-homomorphism on N, then F = τ. Finally, an example is given to demonstrate that the hypothesis of primeness in the above results is essential. [ABSTRACT FROM AUTHOR]