On Jordan Left Derivations of Lie Ideals in Prime Rings

Ashraf, Mohammad; Rehman, Nadeem ur; Ali, Shakir

doi:10.1007/s10012-001-0379-4

Cited by 33 publications

(44 citation statements)

References 4 publications

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“…is an additive mapping such that ) ( Ashraf and Rehman (2000) studied Lie ideals and Jordan left derivations of prime rings. Halder and Paul (2012) extended the results of Ceven (2002) to Lie ideals.…”

Section: Introductionmentioning

confidence: 99%

Generalized (U, M)-derivations of completely semiprime r-rings

Rahman¹,

Paul

2015

J Bangladesh Acad Sci

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Section: Introductionmentioning

confidence: 99%

Generalized (U, M)-derivations of completely semiprime r-rings

Rahman¹,

Paul

2015

J Bangladesh Acad Sci

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“…This result was generalized by many authors (cf. [1], [2], [3] &; [5]). In the present paper, our objective is to generalize this result for derivations defined on a subset of a prime rings.…”

Section: An Additive Mapping D : R -• R Is Called a Derivation If D(xmentioning

confidence: 99%

On Jordan Ideals and Jordan Derivations of Prime Rings

Ashraf¹,

Rehman²

2004

Demonstratio Mathematica

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Abstract. Let R be a 2-torsion free prime ring, and let J be a nonzero Jordan ideal and a subring of R. In the present paper it is shown that if d is an additive mapping of IntroductionThroughout this paper R will denote an associative ring with center Z(R). Recall that R is prime if aRb -(0) implies that a = 0 or b = 0. As usual [x, y] and x o y will denote the commutator xy -yx and anticommutator xy + yx, respectively. A ring R is said to be 2-torsion free, if whenever 2x = 0, with x G R, then x = 0. An additive subgroup J of R is said to be a Jordan ideal of R if u o r € J, for all u G J, r € R. An additive mapping d : R -• R is called a derivation if d(xy) = d(x)y + xd(y), holds for all pairs x, y G R. An additive mapping d : R -> R is called a Jordan derivation if d(x 2 ) = d(x)x + xd(x)holds for all x G R. Obviously, every derivation on a ring R is a Jordan derivation. The converse is, in general, not true. A well known result due to Herstein [6] shows that every Jordan derivation on a 2-torsion free prime ring is a derivation. A brief proof of this result is presented in [4]. This result was generalized by many authors (cf. [5]). In the present paper, our objective is to generalize this result for derivations defined on a subset of a prime rings. Preliminary resultsTo facilitate our discussion, we introduce abbreviation ¿(x, y) = d(xy) -d(x)y -xd(y). We shall make use of commutator identities:

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“…It is easy to prove that g : R → R is a generalized left derivation if and only if g is of the form g = λ + d, where λ : R → R is a right centralizer and d : R → R is a left derivation. If R contains a unit element, then it is easy to see that g is a of the form g = µ k + d, where µ k is a right multiplication by k = λ (1). We here apply the Brešar and Mathieu's result above to arbitrary spectrally bounded generalized left derivations.…”

Section: Spectrally Boundedness Of Generalized Left Derivations and Gmentioning

confidence: 97%

“…d(ab) = ad(b) + bd(a) for all a, b ∈ R). Bresar, Vukman ([7], [17]), Deng [8] and Ashraf et al [1] studied left Jordan derivations and left derivations on prime rings and semiprime rings, which are in a close connection with so-called commuting mappings. Now let us introduce some principal results concerning derivations and related mappings in Banach algebra theory.…”

Section: Introductionmentioning

confidence: 99%

Left Jordan Derivations on Banach Algebras and Related Mappings

Jung¹,

Park²

2010

Bulletin of the Korean Mathematical Society

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Abstract. In this note, we obtain range inclusion results for left Jordan derivations on Banach algebras: (i) Let δ be a spectrally bounded left Jordan derivation on a Banach algebra A. Then δ maps A into its Jacobson radical. (ii) Let δ be a left Jordan derivation on a unital Banach algebra A with the condition sup{r(c −1 δ(c)) : c ∈ A invertible} < ∞. Then δ maps A into its Jacobson radical.Moreover, we give an exact answer to the conjecture raised by Ashraf and Ali in [2, p. 260]: every generalized left Jordan derivation on 2-torsion free semiprime rings is a generalized left derivation.

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On Jordan Left Derivations of Lie Ideals in Prime Rings

Cited by 33 publications

References 4 publications

Generalized (U, M)-derivations of completely semiprime r-rings

Generalized (U, M)-derivations of completely semiprime r-rings

On Jordan Ideals and Jordan Derivations of Prime Rings

Left Jordan Derivations on Banach Algebras and Related Mappings

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