Emine Koç

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.

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 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.

-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 rings to generalized derivations for ()-Lie ideals.