Glasnik Matematicki, Vol. 47, No. 2 (2012), 307-324.

COMPOSITION OF GENERALIZED DERIVATIONS AS A LIE DERIVATION

Vincenzo De Filippis and Giovanni Scudo

Department of Mathematics, University of Messina, 98166 Messina, Italy
e-mail: gscudo@unime.it

Abstract.   Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, F and G non-zero generalized derivations of R. If the composition (FG) acts as a Lie derivation on R, then (FG) is a derivation of R and one of the following holds:

1. there exist α C and a U such that F(x)=[a,x] and G(x)=α x, for all x R;
2. G is an usual derivation of R and there exists α C such that F(x)=α x, for all x R;
3. there exist α, β C and a derivation h of R such that F(x)=α x+h(x), G(x)=β x, for all x R, and α β+h(β)=0. Moreover in this case h is not an inner derivation of R;
4. there exist a', c' U such that F(x)=a'x, G(x)=c'x, for all x R, with a'c'=0;
5. there exist b', q' U such that F(x)=xb', G(x)=xq', for all x R, with q'b'=0;
6. there exist c', q' U, η, γ C such that F(x)=η (xq'-c'x)+γ x, G(x)=c'x+xq', for all x R, with γ c'-η c'²=-γ q'-η q'².

2010 Mathematics Subject Classification.   16N60, 16W25.

Key words and phrases.   Prime rings, differential identities, generalized derivations.

DOI: 10.3336/gm.47.2.07

References:

1. K. I. Beidar, W. S. Martindale and A. V. Mikhalev, Rings with generalized identities, Marcel Dekker, Inc., New York, 1996.
   MathSciNet    

2. M. Brešar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), 525-546.
   MathSciNet     CrossRef

3. M. Brešar, On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J. 33 (1991), 89-93.
   MathSciNet     CrossRef

4. C. L. Chuang, GPIs' having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), 723-728.
   MathSciNet     CrossRef

5. V. De Filippis, Product of two generalized derivations on polynomials in prime rings, Collect. Math. 61 (2010), 303-322.
   MathSciNet     CrossRef

6. V. De Filippis, Generalized derivations as Jordan homomorphisms on lie ideals and right ideals, Acta Math. Sinica (Engl. Ser.) 25 (2009) 1965-1974.
   MathSciNet     CrossRef

7. M. Fošner and J. Vukman, Identities with generalized derivations in prime rings, Mediterranean J. Math. 9 (2012), 847-863.
   MathSciNet     CrossRef

8. V. K. Harčenko, Differential identities of prime rings, Algebra and Logic 17 (1978), 155-168.
   MathSciNet     CrossRef

9. B. Hvala, Generalized derivations in rings, Commun. Algebra 26 (1998), 1147-1166.
   MathSciNet     CrossRef

10. C. Lanski, Differential identities, Lie ideals and Posner's theorems, Pacific J. Math. 134 (1988), 275-297.
    MathSciNet     CrossRef

11. T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27 (1999), 4057-4073.
    MathSciNet     CrossRef

12. J. Ma, X. W. Xu and F. W. Niu, Strong commutativity-preserving generalized derivations on semiprime rings, Acta Math. Sinica (Engl. Ser.) 24 (2008), 1835-1842.
    MathSciNet     CrossRef

13. E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100.
    MathSciNet     CrossRef

14. J. Vukman, On α-derivations of prime and semiprime rings, Demonstratio Math. 38 (2005), 283-290.
    MathSciNet    

15. J. Vukman, Identities related to derivations and centralizers on standard operator algebras, Taiwanese J. Math. 11 (2007), 255-265.
    MathSciNet    

16. J. Vukman, A note on generalized derivations of semiprime rings, Taiwanese J. Math. 11 (2007), 367-370.
    MathSciNet