Central European Journal of Mathematics

, Volume 10, Issue 2, pp 824–834

K-quasiderivations

Research Article
  • 53 Downloads

Abstract

A K-quasiderivation is a map which satisfies both the Product Rule and the Chain Rule. In this paper, we discuss several interesting families of K-quasiderivations. We first classify all K-quasiderivations on the ring of polynomials in one variable over an arbitrary commutative ring R with unity, thereby extending a previous result. In particular, we show that any such K-quasiderivation must be linear over R. We then discuss two previously undiscovered collections of (mostly) nonlinear K-quasiderivations on the set of functions defined on some subset of a field. Over the reals, our constructions yield a one-parameter family of K-quasiderivations which includes the ordinary derivative as a special case.

Keywords

K-quasiderivation Polynomial ring Derivation system 

MSC

13A99 08A40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Adler I., Composition rings, Duke Math. J., 1962, 29(4), 607–623MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Barbeau E.J., Remarks on an arithmetic derivative, Canad. Math. Bull., 1961, 4, 117–122MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Emmons C., Krebs M., Shaheen A., How to differentiate an integer modulo n, College Math. J., 2009, 40(5), 345–353MathSciNetCrossRefGoogle Scholar
  4. [4]
    Fechter T., Exploring the Derivative of a Natural Number Using the Logarithmic Derivative, Senior Capstone thesis, Pacific University, 2007Google Scholar
  5. [5]
    Gleason A.M., Greenwood R.E., Kelly L.M. (Eds.), The William Lowell Putnam Mathematical Competition. Problems and Solutions: 1938–1964, Mathematical Association of America, Washington, 1980MATHGoogle Scholar
  6. [6]
    Kautschitsch H., Müller W.B., Über die Kettenregel in A [x 1,...x n], A (x 1...x n) und A [[x 1,...x n]], In: Contributions to General Algebra, 1, Klagenfurt, May 25–28, 1978, Johannes Heyn, Klagenfurt, 1979, 131–136Google Scholar
  7. [7]
    Lausch H., Nöbauer W., Algebra of Polynomials, North-Holland Math. Library, 5, North-Holland, Amsterdam-London, 1973Google Scholar
  8. [8]
    Menger K., General algebra of analysis, Reports of Mathematical Colloquium, 1946, 7, 46–60MathSciNetGoogle Scholar
  9. [9]
    Müller W., Eindeutige Abbildungen mit Summen-, Produkt- und Kettenregel im Polynomring, Monatsh. Math., 1969, 73(4), 354–367MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Müller W.B., The algebra of derivations, An. Acad. Brasil. Ciênc., 1973, 45, 339–343 (in Spanish)Google Scholar
  11. [11]
    Müller W.B., Differentiations-Kompositionsringe, Acta Sci. Math. (Szeged), 1978, 40(1–2), 157–161MathSciNetMATHGoogle Scholar
  12. [12]
    Müller W.B., Über die Kettenregel in Fastringen, Abh. Math. Sem. Univ. Hamburg, 1979, 48(1), 108–111MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Nöbauer W., Derivationssysteme mit Kettenregel, Monatsh. Math., 1963, 67(1), 36–49MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Stay M., Generalized number derivatives, J. Integer Seq., 2005, 8(1), #05.1.4Google Scholar
  15. [15]
    Ufnarovski V., Ahlander B., How to differentiate a number, J. Integer Seq., 2003, 6(3), #03.3.4Google Scholar
  16. [16]
    Westrick L., Investigations of the number derivative, preprint available at http://www.plouffe.fr/simon/OEIS/archive_in_pdf/intmain.pdf

Copyright information

© © Versita Warsaw and Springer-Verlag Wien 2011

Authors and Affiliations

  1. 1.Department of MathematicsCalifornia State University - Los AngelesLos AngelesUSA

Personalised recommendations