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On Generalized Jordan Left Derivations in Prime Γ–Rings

February 2018

Authors:

Neshtiman N. Sulaiman

Salahaddin University-Erbil

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ResearchGate has not been able to resolve any references for this publication.

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September 2004 · International Journal of Mathematics and Mathematical Sciences

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In a recent paper we have extended the classical Herstein's theorem on Jordan derivations on prime rings to Jordan superderivations on prime associative superalgebras. In the present paper we extend this result to semiprime associative superalgebras.

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The purpose of this paper is to study derivations and commutativity identities for a prime near ring. Suppose G 1 is a prime near ring, N is the set of natural numbers and let D be derivation on G 1 then, G 1 is said to be commutative ring if there exist α, β ∈ N such that D([t, f ]) = f α [t, f ]f β or D([t, f ]) = −f α [t, f ]f β for all t, f ∈ G 1. Furthermore, D(t • f) = f α (t • f)f β or D(t ... [Show full abstract] • f) = −f α (t • f)f β for all t, f ∈ G 1. Where [t, f ] and (t • f) are lie and Jordan products respectively.

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L.O. Chung

The notion of nil derivations is a generalization of the notion of nilpotent derivations. The latter, because of its close relation with automorphisms and the existence of a Jordan decomposition into semisimple and nilpotent parts for a large family of derivations, has received considerable attention recently. For a prime ring of characteristic 0, a relation between a nil derivation being inner ... [Show full abstract] with the existence of nontrivial fixed points of its corresponding automorphism is established. From this, a criterion on ∂ being “inner” and induced by a nil element is derived. As an application, the result that a nilpotent derivation is induced by a nilpotent element in the endomorphism ring End(IR, IR), where IR is certain ideal of R is deduced. This is a generalization of some well-known results due to Kharchenko and others.

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Let R be a prime ring with center Z(R)Z(R) and with extended centroid C . We give a complete characterization of Jordan derivations of R when charR=2 and dimC⁡RC=4dimC⁡RC=4: An additive map δ:R→RCδ:R→RC is a Jordan derivation if and only if there exist a derivation d:R→RCd:R→RC and an additive map μ:R→Cμ:R→C such that δ=d+μδ=d+μ and μ(x2)=0μ(x2)=0 for all x∈Rx∈R. As consequences, it is proved ... [Show full abstract] among other things: Any Z(R)Z(R)-linear Jordan derivation of R is a derivation if dimC⁡RC<∞dimC⁡RC<∞. Moreover, if C is either a finite field or an algebraically closed field, where charC=2 and n≥2n≥2, then every Jordan derivation of Mn(C)Mn(C) is a derivation.

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In this article, we develop some important results relating to the concepts of generalized k -derivation and Jordan generalized k-derivation of certain gamma rings. Though every generalized k-derivation of a gamma ring M is obviously a Jordan generalized k-derivation of M, but the converse statement is in general not true. Here we prove that every Jordan generalized k-derivation of a 2-torsion ... [Show full abstract] free prime gamma ring (in the sense of Nobusawa) is a generalized k-derivation.

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Jordan derivations of prime rings1

January 1957

I.N. Herstein

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Let X be a real or complex Banach space, L(X) be the algebra of all bounded linear operators of X into itself and A(X) � L(X) be a standard oper- ator algebra. Suppose there exist linear mappings D;G: A(X) ! L(X) satisfying the relations 2D(An) = D(An􀀀1)A + An􀀀1G(A) + G(A)An􀀀1 + AG(An􀀀1) and 2G(An) = G(An􀀀1)A + An􀀀1D(A) + D(A)An􀀀1 + AD(An􀀀1) for all A 2 A(X). Then there exists a �xed B 2 L(X) ... [Show full abstract] such that D(A) = G(A) = [A;B] holds for all A 2 A(X).

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In this paper we extend our ideas from reverse derivation towards the Generalized reverse derivations on semiprime rings. In this Paper, we prove that if d is a non-zero reverse derivation of a semi prime ring R and f is a generalized reverse derivation, then �� is a strong commutativity preserving. Using this, we prove that R is commutative.

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JORDAN AND LIE HOMOMORPHISM AND DEVIRATION ON PRIME RING

January 1970

REBECCA LUO

We will discuss some properties of Jordan and Lie homomorphisms and derivations on prime rings. We first give some definitions and then we show some classic results .

On Generalized Jordan Left Derivations in Prime Γ–Rings

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