On Generalized (Sigma,Tau)-Derivations in 3-Prime Near-Rings

Let N be a 3-prime left near-ring with multiplicative center Z, f be a generalized (σ,τ)- derivation on N with associated (σ,τ)-derivation d and I be a semigroup ideal of N. We proved that N must be a commutative ring if f(I)⊂Z or f act as a homomorphism or f act as an anti-homomorphism.

Semigroup ideals and generalized semiderivations of prime near rings

Let N be a near ring. An additive mapping F : N 􀀀! N is said to be a generalized semiderivation on N if there exists a semiderivation d : N 􀀀! N associated with a function g : N 􀀀! N such that F(xy) = F(x)y + g(x)d(y) = d(x)g(y) + xF(y) and F(g(x)) = g(F(x)) for all x; y 2 N. In this paper we prove some theorems in the setting of semigroup ideal of a 3-prime near ring admitting a nonzero generalized semiderivation associated with a nonzero semiderivation, thereby extending some known results on derivations, semiderivations and generalized derivations.

HUBUNGAN DERIVASI PRIME NEAR-RING DENGAN SIFAT KOMUTATIF RING

Near-rings are generalize from rings. A research on near-ring is continous included a research on prime near-rings and one of this research is about derivation on prime near-rings. This article will reviewing about relation between derivation on prime near-rings and commutativity in rings with literature review method. The result is prime near-rings are commutative rings if a nonzero derivation d on N hold one of this following conditions: (i) , (ii) , (iii) , (iv) , (v) , (vi) , for all , with is non zero semigroup ideal from .

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

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).

Generalized Derivations on Prime Near Rings

LetNbe a near ring. An additive mappingf:N→Nis said to be a right generalized (resp., left generalized) derivation with associated derivationdonNiff(xy)=f(x)y+xd(y)(resp.,f(xy)=d(x)y+xf(y)) for allx,y∈N. A mappingf:N→Nis said to be a generalized derivation with associated derivationdonNiffis both a right generalized and a left generalized derivation with associated derivationdonN. The purpose of the present paper is to prove some theorems in the setting of a semigroup ideal of a 3-prime near ring admitting a generalized derivation, thereby extending some known results on derivations.

$(\theta,\phi)$-derivations as homomorphisms or as anti-homomorphisms on a near ring

Let $N$ be a near ring. An additive mapping $d:N\longrightarrow N$ is said to be a $(\theta,\phi)$-derivation on $N$ if there exist mappings $\theta,\phi:N\longrightarrow N$ such that$d(xy)=\theta(x)d(y)+d(x)\phi(y)$ holds for all $x,y \in N$. In the context of 3-prime and 3-semiprime nearrings, we show that for suitably-restricted $\theta$ and $\phi$, there exist no nonzero $(\theta,\phi)$-derivations which act as a homomorphism or an anti-homomorphism on $N$ or a nonzero semigroup ideal of $N$.

Semigroup ideal in Prime Near-Rings with Derivations

In this paper we generalize some of the results due to Bell and Mason on a near-ring N admitting a derivation D , and we will show that the body of evidence on prime near-rings with derivations have the behavior of the ring. Our purpose in this work is to explore further this ring like behavior. Also, we show that under appropriate additional hypothesis a near-ring must be a commutative ring.

Semigroup ideals in rings

A ringRis called anl-ring (r-ring) in caseRcontains an indentity and every left (right) semigroup ideal is a left (right) ring ideal. A number of structure theorems are obtained forl-rings whenRis left noetherian and left artinian. It is shown that left noetherianl-rings are local left principal ideal rings. WhenRis a finite dimensional algebra over a field, the property of being anl-ring is equivalent to being anr-ring. However, examples are given to show that these two concepts are in general not equivalent even in the artinian case.

Semigroups in rings

A subset S of a ring R is a left semigroup ideal of R if RS ⊈ R, and a left ring ideal of R if in addition S is a subring of R. Gluskin (1960) investigated those rings with 1 which possess the property: (λ) every left semigroup ideal is a left ring ideal.