Oznur Golbasi

On Prime and Semiprime Near-Rings with Generalized Derivation

Abstract Let N be a left near-ring and S be a nonempty subset of N. A mapping F from N to N is called commuting on S if [F(x),x] = 0 for all x € S. The mapping F is called strong commutativity preserving (SCP) on S if [F(x),F(y)] = [x,y] for all x, y € S. In the present paper, firstly we generalize the well known result of Posner which is commuting derivations on prime rings to generalized derivations of semiprime near-rings. Secondly, we investigate SCP-generalized derivations of prime near-rings.

SOME COMMUTATIVITY THEOREMS OF PRIME RINGS WITH GENERALIZED (σ, τ)-DERIVATION

In this paper, we extend some well known results concerning generalized derivations of prime rings to a generalized (; )-derivation.

Some properties of prime near-rings with (σ,τ)-derivation

Some results by Bell and Mason on commutativity in near-rings are generalized. Let N be a prime right near-ring with multiplicative center Z and let D be a (σ,τ)-derivation on N such that σD = Dσ and τD = Dτ. The following results are proved: (i) If D(N) ⊂ Z or [D(N), D(N)] = 0 or [D(N), D(N)]σ,τ = 0 then (N, +) is abelian; (ii) If D(xy) = D(x)D(y) or D(xy) = D(y)D(x) for all x, y ∈ N then D = 0.

Notes on Generalized

Let R be a prime ring with charR not equal 2 and let sigma, tau be automorphisms of R. An additive mapping f : R -> R is called a generalized (sigma, tau)-derivation if there exists a (sigma, tau)-derivation d : R -> R such that f(xy) = f(x)sigma(y)+tau(x)d(y) holds for all x,y is an element of R. In this paper, some well known results concerning generalized derivations of prime rings are extended to generalized (sigma,tau)-derivations.

(\sigma,\tau)

Abstract Let R be a ∗−prime ring with characteristic not 2, U a nonzero ∗− (σ,τ)−Lie ideal of R, d a nonzero derivation of R. Suppose σ, τ be two automorphisms of R such that σd = dσ, τd = dτ and ∗ commutes with σ, τ, d. In the present paper it is shown that if d(U) ⊆ Z or d2(U) ⊆ Z, then U ⊆ Z.

-Derivation

ABSTRACT Let R be a semiprime ring and I a nonzero ideal of R. A map F:R→R is called a multiplicative generalized derivation if there exists a map d:R→R such that F(xy) = F(x)y+xd(y), for all x,y∈R. In the present paper, we shall prove that R contains a nonzero central ideal if any one of the following holds: i) ii) iii) F is SCP on I, iv) F(u)∘F(v) = u∘v, for all u,v∈I.

Remarks on *−(σ,Τ)− Lie Ideals of *−Prime Rings with Derivation

Abstract Let N be a semiprime right near-ring and I a semigroup ideal of N. A map f : N → N is called a multiplicative generalized (θ, θ)–derivation if there exists a multiplicative (θ, θ)–derivation d : R → R such that f(xy) = f(x)θ(y) + θ(x)d(y), for all x, y ∈ R. The purpose of this paper is to investigate the following: (i) f(uv) = ±uv, (ii) f(uv) = ±vu, (iii) f(u)f(v) = ±uv, (iv) f(u)f(v) = ±vu, (v) d(u)d(v) = θ ([u, v]), (vi) d(u)d(v) = θ (uov), (vii) d(u)θ(v) = θ(u)d(v).

Some results on ideals of semiprime rings with multiplicative generalized derivations

We extend some well known commutativity results concerning a nonzero square closed -Lie ideal and generalized -derivations of -prime rings.

Semiprime Near-Rings with Multiplicative Generalized (θ, θ)–Derivations

In the present paper, we extend some well known results concerning derivations of prime near-rings in [4], [5] and [13] to (σ, τ)−derivations and semigroup ideals of prime near-rings. 2000 AMS Classification: 16Y30, 16W25.

Some Results on Generalized -Derivations in-Prime Rings

Let R be a prime ring with center Z and characteristic different from two, U a nonzero Lie ideal of R such that for all and be a nonzero (, )-derivation of R. We prove the following results: (i) If = 0 or for all , then . (ii) If and = 0 for all , then or . (iii) If for all , then .