Generalized Derivations on Lie Ideals and Power Values on Prime Rings
-
Giovanni Scudo
and
Abu Zaid Ansari
Abstract
Let R be a non-commutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, L a non-central Lie ideal of R, G a non-zero generalized derivation of R.
If [G(u), u]n = [G(u), u], for all u ∈ L, with n > 1, then one of the following holds:
(1) R satisfies the standard identity s4(x1, . . . , x4) and there exist a ∈ U and α ∈ C such that G(x) = ax + xa + αx for all x ∈ R;
(2) there exists γ ∈ C such that G(x) = γx for all x ∈ R.
References
[1] BEIDAR, K. I.: Rings with generalized identities, Moscow Univ. Math. Bull. 33 (1978), 53-58.Search in Google Scholar
[2] BEIDAR, K. I.-MARTINDALE, W. S.-MIKHALEV, A. V.: Rings with generalized identities. Pure Appl. Math., Dekker, New York, 1996.Search in Google Scholar
[3] CARINI, L.-DE FILIPPIS, V.: Commutators with power central values on a Lie ideal, Pacific J. Math. 193 (2000), 269-278.10.2140/pjm.2000.193.269Search in Google Scholar
[4] CHUANG, C. L.: GPIs’ having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), 723-728.10.1090/S0002-9939-1988-0947646-4Search in Google Scholar
[5] DE FILIPPIS, V.: Generalized derivations and commutators with nilpotent values on Lie ideals, Tamsui Oxf. J. Math. Sci. 22 (2006), 167-175.Search in Google Scholar
[6] DI VINCENZO, O. M.: On the n-th centralizer of a Lie ideal, Boll. Unione Mat. Ital. 7 (1989), 77-85.Search in Google Scholar
[7] ERICSON, T. S.-MARTINDALE III, W. S.-OSBORN, J. M.: Prime nonassociative algebras, Pacific J. Math. 60 (1975), 49-63.10.2140/pjm.1975.60.49Search in Google Scholar
[8] HERSTEIN, I. N.: Topics in Ring Theory, Univ. Chicago Press, Chicago, IL, 1969.Search in Google Scholar
[9] JACOBSON, N.: Structure of Rings, Amer. Math. Soc., Providence, RI, 1964.Search in Google Scholar
[10] KHARCHENKO, V. K.: Differential identities of prime rings, Algebra Logic 17 (1978), 155-168.10.1007/BF01670115Search in Google Scholar
[11] LANSKI, C.: Derivations with nilpotent values on Lie ideals, Proc. Amer. Math. Soc. 108 (1990), 31-37.10.1090/S0002-9939-1990-0984803-4Search in Google Scholar
[12] LEE, T. K.: Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sin. 20 (1992), 27-38.Search in Google Scholar
[13] LEE, T. K.: Generalized derivations of left faithful rings, Comm. Algebra 27 (1999), 4057-4073.10.1080/00927879908826682Search in Google Scholar
[14] MARTINDALE III, W. S.: Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576-584.10.1016/0021-8693(69)90029-5Search in Google Scholar
[15] POSNER, E. C.: Derivations in prime rings, Proc. Amer.Math. Soc. 8 (1957), 1093-1100.10.1090/S0002-9939-1957-0095863-0Search in Google Scholar
Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Finite Mixed Sums wih Harmonic Terms
- Packing of ℝ2 by Crosses
- On the Integrality of the Elementary Symmetric Functions of 1, 1/3, . . . , 1/(2n − 1)
- Generalized Derivations as a Generalization of Jordan Homomorphisms Acting on Lie Ideals and Right Ideals
- Generalized Derivations on Lie Ideals and Power Values on Prime Rings
- On Monoids of Injective Partial Cofinite Selfmaps
- Extensions of Dynamic Inequalities of Hardy’s Type on Time Scales
- The Controlled Convergence Theorem for the Gap-Integral
- The Solvability of a Nonlocal Boundary Value Problem
- Oscillation Criteria for Third Order Differential Equations with Functional Arguments
- Asymptotic Behavior of Solutions of a Nonlinear Neutral Generalized Pantograph Equation with Impulses
- On Null Lagrangians
- Principal Eigenvalues for Systems of Schrödinger Equations Defined in the whole Space with Indefinite Weights
- Convergence of Series on Large Set of Indices
- On Approximation Properties of a New Type of Bernstein-Durrmeyer Operators
- Representation of Extendible Bilinear Forms
- Spectra and Fine Spectra of Lower Triangular Double-Band Matrices as Operators on Lp (1 ≤ p < ∞)
- Topological Fundamental Groups and Small Generated Coverings
- A Relation between two Kinds of Norms for Martingales
- Linearization Regions in Singular Weakly Nonlinear Regression Models with Constraints
- Parametric Equilibrium Problems Governed by Topologically Pseudomonotone Bifunctions
- Identification of a Parameter in Fourth-Order Partial Differential Equations by an Equation Error Approach
Articles in the same Issue
- Finite Mixed Sums wih Harmonic Terms
- Packing of ℝ2 by Crosses
- On the Integrality of the Elementary Symmetric Functions of 1, 1/3, . . . , 1/(2n − 1)
- Generalized Derivations as a Generalization of Jordan Homomorphisms Acting on Lie Ideals and Right Ideals
- Generalized Derivations on Lie Ideals and Power Values on Prime Rings
- On Monoids of Injective Partial Cofinite Selfmaps
- Extensions of Dynamic Inequalities of Hardy’s Type on Time Scales
- The Controlled Convergence Theorem for the Gap-Integral
- The Solvability of a Nonlocal Boundary Value Problem
- Oscillation Criteria for Third Order Differential Equations with Functional Arguments
- Asymptotic Behavior of Solutions of a Nonlinear Neutral Generalized Pantograph Equation with Impulses
- On Null Lagrangians
- Principal Eigenvalues for Systems of Schrödinger Equations Defined in the whole Space with Indefinite Weights
- Convergence of Series on Large Set of Indices
- On Approximation Properties of a New Type of Bernstein-Durrmeyer Operators
- Representation of Extendible Bilinear Forms
- Spectra and Fine Spectra of Lower Triangular Double-Band Matrices as Operators on Lp (1 ≤ p < ∞)
- Topological Fundamental Groups and Small Generated Coverings
- A Relation between two Kinds of Norms for Martingales
- Linearization Regions in Singular Weakly Nonlinear Regression Models with Constraints
- Parametric Equilibrium Problems Governed by Topologically Pseudomonotone Bifunctions
- Identification of a Parameter in Fourth-Order Partial Differential Equations by an Equation Error Approach
