What is a Functional Identity?

Part of the Frontiers in Mathematics book series (FM)


An exposition on a mathematical subject usually starts with basic definitions. We feel, however, that it is more appropriate to introduce functional identities through examples. We shall therefore present various simple examples of functional identities, so that the reader may guess which conclusions could be derived when facing these identities. These examples have been selected in order to illustrate the general theory, and not all of them are of great importance in their own right. Their consideration will be rather elementary; anyhow, many of the arguments that we shall present here will be used, sometimes in a hidden way, in much more general situations considered in further chapters. Examples will be followed by some basic definitions and notation, but even these will be given in a somewhat informal fashion. The last objective of this preliminary chapter is to point out a few instances where functional identities appear naturally. That is, we wish to indicate, without many details, why and where the theory of functional identities is applicable. So, in summary, the goal of this chapter is to give an informal introduction to functional identities which should be of help to a newcomer to the subject.


Prime Ring Division Ring Associative Ring Functional Identity Unital Ring 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature and Comments

  1. [6]
    P. Ara, M. Mathieu, An application of local multipliers to centralizing mappings of C*-algebras, Quart. J. Math. 44 (1993), 129–138.MATHCrossRefMathSciNetGoogle Scholar
  2. [12]
    R. Banning, M. Mathieu, Commutativity preserving mappings on semiprime rings, Comm. Algebra 25 (1997), 247–265.MATHCrossRefMathSciNetGoogle Scholar
  3. [16]
    K. I. Beidar, On functional identities and commuting additive mappings, Comm. Algebra 26 (1998), 1819–1850.MATHCrossRefMathSciNetGoogle Scholar
  4. [29]
    K. I. Beidar, M.A. Chebotar, On functional identities and d-free subsets of rings I, Comm. Algebra 28 (2000), 3925–3951.MATHCrossRefMathSciNetGoogle Scholar
  5. [35]
    K. I. Beidar, Y. Fong, P.-H. Lee, T.-L. Wong, On additive maps of prime rings satisfying Engel condition, Comm. Algebra 25 (1997), 3889–3902.MATHCrossRefMathSciNetGoogle Scholar
  6. [39]
    K. I. Beidar, W. S. Martindale 3rd, A.V. Mikhalev, Lie isomorphisms in prime rings with involution, J. Algebra 169 (1994), 304–327.MATHCrossRefMathSciNetGoogle Scholar
  7. [54]
    M. Brešar, Centralizing mappings on von Neumann algebras, Proc. Amer. Math. Soc. 111 (1991), 501–510.CrossRefMATHMathSciNetGoogle Scholar
  8. [55]
    M. Brešar, On a generalization of the notion of centralizing mappings, Proc. Amer. Math. Soc. 114 (1992), 641–649.CrossRefMATHMathSciNetGoogle Scholar
  9. [56]
    M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993), 385–394.CrossRefMATHMathSciNetGoogle Scholar
  10. [57]
    M. Brešar, On skew-commuting mappings of rings, Bull. Austral. Math. Soc. 47 (1993), 291–296.MATHMathSciNetGoogle Scholar
  11. [58]
    M. Brešar, Commuting traces of biadditive mappings, commutativitypreserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), 525–546.CrossRefMATHMathSciNetGoogle Scholar
  12. [59]
    M. Brešar, On certain pairs of functions of semiprime rings, Proc. Amer. Math. Soc. 120 (1994), 709–713.CrossRefMATHMathSciNetGoogle Scholar
  13. [60]
    M. Brešar, On generalized biderivations and related maps, J. Algebra 172 (1995), 764–786.CrossRefMATHMathSciNetGoogle Scholar
  14. [62]
    M. Brešar, Applying the theorem on functional identities, Nova Journal of Mathematics, Game Theory, and Algebra 4 (1995), 43–54.Google Scholar
  15. [63]
    M. Brešar, On a certain identity satisfied by a derivation and an arbitrary additive mapping II, Aequationes Math. 51 (1996), 83–85.CrossRefMATHMathSciNetGoogle Scholar
  16. [64]
    M. Brešar, Functional identities: A survey, Contemporary Math. 259 (2000), 93–109.Google Scholar
  17. [66]
    M. Brešar, Commuting maps: A survey, Taiwanese J. Math. 8 (2004), 361–397.MATHMathSciNetGoogle Scholar
  18. [74]
    M. Brešar, B. Hvala, On additive maps of prime rings, Bull. Austral. Math. Soc. 51 (1995), 377–381.CrossRefMATHMathSciNetGoogle Scholar
  19. [75]
    M. Brešar, B. Hvala, On additive maps of prime rings, II, Publ. Math. Debrecen 54 (1999), 39–54.MATHMathSciNetGoogle Scholar
  20. [77]
    M. Brešar, W. S. Martindale 3rd, C. R. Miers, Centralizing maps in prime rings with involution, J. Algebra 161 (1993), 342–357.CrossRefMATHMathSciNetGoogle Scholar
  21. [78]
    M. Brešar, W. S. Martindale 3rd, C. R. Miers, Maps preserving n th powers, Comm. Algebra 26 (1998), 117–138.CrossRefMATHMathSciNetGoogle Scholar
  22. [79]
    M. Brešar, C. R. Miers, Commutativity preserving mappings of von Neumann algebras, Canad. J. Math. 45 (1993), 695–708.MATHMathSciNetGoogle Scholar
  23. [80]
    M. Brešar, C.R. Miers, Strong commutativity preserving maps of semiprime rings, Canad. Math. Bull. 37 (1994), 457–460.MATHMathSciNetGoogle Scholar
  24. [81]
    M. Brešar, C. R. Miers, Commuting maps on Lie ideals, Comm. Algebra 23 (1995), 5539–5553.CrossRefMATHMathSciNetGoogle Scholar
  25. [87]
    M. A. Chebotar, On Lie automorphisms of simple rings of characteristic 2, Fundam. Prikl. Mat. 2 (1996), 1257–1268 (Russian).MATHMathSciNetGoogle Scholar
  26. [88]
    M. A. Chebotar, On generalized functional identities in prime rings, J. Algebra 202 (1998), 655–670.MATHCrossRefMathSciNetGoogle Scholar
  27. [107]
    D. R. Farkas, G. Letzter, Ring theory from symplectic geometry, J. Pure Appl. Algebra 125 (1998), 155–190.MATHCrossRefMathSciNetGoogle Scholar
  28. [137]
    T.-C. Lee, Derivations and centralizing maps on skew elements, Soochow J. Math. 24 (1998), 273–290.MATHMathSciNetGoogle Scholar
  29. [139]
    P.-H. Lee, T.-K. Lee, Linear identities and commuting maps in rings with involution, Comm. Algebra 25 (1997), 2881–2895.MATHCrossRefMathSciNetGoogle Scholar
  30. [140]
    P.-H. Lee, J.-S. Lin, R.-J. Wang, T.-L. Wong, Commuting traces of multiadditive mappings, J. Algebra 193 (1997), 709–723.MATHCrossRefMathSciNetGoogle Scholar
  31. [141]
    T.-K. Lee, σ-commuting mappings in semiprime rings, Comm. Algebra 29 (2001), 2945–2951.MATHCrossRefMathSciNetGoogle Scholar
  32. [142]
    T.-K. Lee, T.-C. Lee, Commuting additive mappings in semiprime rings, Bull. Inst. Math. Acad. Sinica 24 (1996), 259–268.MATHMathSciNetGoogle Scholar
  33. [159]
    J. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19 (1976), 113–115.MATHMathSciNetGoogle Scholar
  34. [182]
    E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100.CrossRefMathSciNetGoogle Scholar
  35. [189]
    V. G. Skosyrskii, Strongly prime noncommutative Jordan algebras, Trudy Inst. Mat. (Novosibirsk) 16 (1989), 131–164 (in Russian).MathSciNetGoogle Scholar
  36. [192]
    G. A. Swain, Lie derivations of the skew elements of prime rings with involution, J. Algebra 184 (1996), 679–704.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag 2007

Personalised recommendations