On commutativity of rings with generalized derivations

In the present paper, we introduce the notion of (σ, τ)-n-derivation in near-ring N and investigate some properties involving (σ, τ)-n-derivations of a prime near-ring N which force N to be a commutative ring. Additive commutativity of near-ring N satisfying certain identities involving (σ, τ)-n-derivations of a prime near-ring N has also been obtained. Related examples to justify the hypotheses in various theorems have also been provided.

ON (σ, τ)-n-DERIVATIONS IN NEAR-RINGS

derivation) on S if δ(xy) = θ(x)δ(y) + φ(y)δ(x) (resp., δ(x 2 ) = θ(x)δ(x)+ φ(x)δ(x)) holds for all x, y ∈ S. Suppose that J is a Jordan ideal and a subring of a 2-torsion-free prime ring R. In the present paper, it is shown that if θ is an automorphism of R such that δ(x 2 ) = 2θ(x)δ(x) holds for all x ∈ J, then either J ⊆ Z(R) or δ(J) = (0). Further, a study of left (θ, θ)-derivations of a prime ring R has been made which acts either as a homomorphism or as an antihomomorphism of the ring R.

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ON JORDAN IDEALS AND LEFT (θ, θ)-DERIVATIONS IN PRIME RINGS

In this paper, we investigate commutativity of ring R with involution ∗ which admits a derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Finally, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous.

Commutativity theorems in rings with involution

Let $R$ be a 2-torsion free semiprime ring. In this paper we will show that every Jordan triple $(\theta,\phi)$-derivation on $R$ is a $(\theta,\phi)$-derivation. Also every Jordan triple left centralizer on $R$ is a left centralizer. As a consequence, every generalized Jordan triple $(\theta,\phi)$-derivation on $R$ is a generalized $(\theta,\phi)$-derivation. This result gives an affirmative answer to the question posed by Wu and Lu in [14].

http://journal.taiwanmathsoc.org.tw/~journal/tjm/V11N5/m11610-liujordantri051210.pdf

GENERALIZED JORDAN TRIPLE (θ, φ)-DERIVATIONS ON SEMIPRIME RINGS

Let R be a ring and d : R → R a derivation of R. In the present paper we investigate commutativity of R satisfying any one of the properties (i)d([x,y]) = [x,y], (ii)d(x o y) = xoy, (iii)d(x) o d(y) = 0, or (iv)d(x) o d(y) = x o y, for all x, y in some apropriate subset of R.

On Commutativity of Rings With Derivations

Let R be a prime ring with characteristic different from two and S a non-empty subset of R. Suppose that θ, Φ are endomorphisms of R. An additive mapping F : R → R is called a generalized (θ, Φ)-derivation (resp. generalized Jordan (θ, Φ)-derivation) on S if there exists a (θ, Φ)-derivation d : R → R such that F(xy) = F(x)θ(y) + Φ(x)d(y) (resp. F(x 2) = F(x)θ(x) + Φ(x)d(x)), holds for all x, y ∈ S. Suppose that U is a Lie ideal of R such that u 2 ∈ U, for all u ∈ U. In the present paper, it is shown if θ is an automorphism of R then every generalized Jordan (θ, Φ)-derivation F on U is a generalized (θ, Φ)-derivation on U.

On Lie Ideals and Generalized ( θ Φ Φ )-Derivations in Prime Rings

Let R be a prime ring with characteristic different from two and U be a Lie ideal of R such that u2 ɛ U for all u ɛ U. In the present paper it is shown that if d is an additive mappings of R into itself satisfying d(u2) = 2ud(u), for all u ɛ U, then either U ɛ Z(R) or d(U) = (0).

On Jordan Left Derivations of Lie Ideals in Prime Rings

There is an increasing body of evidence that prime near-rings with derivations have ring like behavior, indeed, there are several results (see for example [1], [2], [3], [4], [5] and [8]) asserting that the existence of a suitably-constrained derivation on a prime near-ring forces the near-ring to be a ring. It is our purpose to explore further this ring like behaviour. In this paper we generalize some of the results due to Bell and Mason [4] on near-rings admitting a special type of derivation namely $(\sigma ,\tau )$- derivation where $\sigma ,\tau $ are automorphisms of the near-ring. Finally, it is shown that under appropriate additional hypothesis a near-ring must be a commutative ring.

$(\sigma ,\tau )$-derivations on prime near rings

Let R be a ring and fi;fl be endomorphisms of R. An additive mapping F : R ! R is called a generalized (fi;fl)-derivation on R if there exists an (fi;fl)-derivation d : R ! R such that F(xy) = F(x)fi(y) + fl(x)d(y) holds for all x;y 2 R. In the present paper, we discuss the commutativity of a prime ring R admitting a generalized (fi;fl)- derivation F satisfying any one of the properties: (i) (F(x);x)fi;fl = 0, (ii) F((x;y)) = 0, (iii) F(x-y) = 0, (iv) F((x;y)) = (x;y)fi;fl, (v) F(x-y) = (x-y)fi;fl, (vi) F(xy)ifi(xy) 2 Z(R), (vii) F(x)F(y) i fi(xy) 2 Z(R) for all x;y in an appropriate subset of R. 2000 Mathematics Subject Classiflcation: 16W25, 16N60, 16U80

Mohammad Ashraf

Papers

On Commutativity of Rings With Derivations

On Lie Ideals and Generalized ( θ Φ Φ )-Derivations in Prime Rings

On Jordan Left Derivations of Lie Ideals in Prime Rings

$(\sigma ,\tau )$-derivations on prime near rings

On Generalized (fi;fl)-Derivations in Prime Rings