K-quasiderivations

• Autorzy: Caleb Emmons , Mike Krebs , Anthony Shaheen
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

K-quasiderivation; Polynomial ring; Derivation system

Kategorie tematyczne

• 08A40: Operations, polynomials, primal algebras
• 13A99: None of the above, but in this section

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 2
• Strony: 824-834

Daty

wydano

2012-04-01

online

2012-01-18

Twórcy

autor

Caleb Emmons

• California State University - Los Angeles

autor

Mike Krebs

• California State University - Los Angeles

autor

Anthony Shaheen

• California State University - Los Angeles

Bibliografia

• [1] Adler I., Composition rings, Duke Math. J., 1962, 29(4), 607–623 http://dx.doi.org/10.1215/S0012-7094-62-02961-7
• [2] Barbeau E.J., Remarks on an arithmetic derivative, Canad. Math. Bull., 1961, 4, 117–122 http://dx.doi.org/10.4153/CMB-1961-013-0
• [3] Emmons C., Krebs M., Shaheen A., How to differentiate an integer modulo n, College Math. J., 2009, 40(5), 345–353 http://dx.doi.org/10.4169/074683409X475661
• [4] Fechter T., Exploring the Derivative of a Natural Number Using the Logarithmic Derivative, Senior Capstone thesis, Pacific University, 2007
• [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, 1980
• [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, 131–136
• [7] Lausch H., Nöbauer W., Algebra of Polynomials, North-Holland Math. Library, 5, North-Holland, Amsterdam-London, 1973
• [8] Menger K., General algebra of analysis, Reports of Mathematical Colloquium, 1946, 7, 46–60
• [9] Müller W., Eindeutige Abbildungen mit Summen-, Produkt- und Kettenregel im Polynomring, Monatsh. Math., 1969, 73(4), 354–367 http://dx.doi.org/10.1007/BF01298986
• [10] Müller W.B., The algebra of derivations, An. Acad. Brasil. Ciênc., 1973, 45, 339–343 (in Spanish)
• [11] Müller W.B., Differentiations-Kompositionsringe, Acta Sci. Math. (Szeged), 1978, 40(1–2), 157–161
• [12] Müller W.B., Über die Kettenregel in Fastringen, Abh. Math. Sem. Univ. Hamburg, 1979, 48(1), 108–111 http://dx.doi.org/10.1007/BF02941293
• [13] Nöbauer W., Derivationssysteme mit Kettenregel, Monatsh. Math., 1963, 67(1), 36–49 http://dx.doi.org/10.1007/BF01300680
• [14] Stay M., Generalized number derivatives, J. Integer Seq., 2005, 8(1), #05.1.4
• [15] Ufnarovski V., Ahlander B., How to differentiate a number, J. Integer Seq., 2003, 6(3), #03.3.4
• [16] Westrick L., Investigations of the number derivative, preprint available at http://www.plouffe.fr/simon/OEIS/archive_in_pdf/intmain.pdf

Identyfikatory

DOI

10.2478/s11533-011-0140-x