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POLYNOMIAL EQUIVALENCE OF FINITE RINGS

Part of: Sequences and sets Finite commutative rings Ring extensions and related topics

Published online by Cambridge University Press:  01 April 2014

GEORG GRASEGGER ,
GÁBOR HORVÁTH  and
KEITH A. KEARNES
GEORG GRASEGGER
Affiliation:
Doctoral Program Computational Mathematics, Research Institute for Symbolic Computation, Johannes Kepler University Linz, Altenberger Strasse 69, 4040 Linz, Austria email Georg.Grasegger@risc.jku.at
GÁBOR HORVÁTH*
Affiliation:
Institute of Mathematics, University of Debrecen, Pf. 12, 4010 Debrecen, Hungary
KEITH A. KEARNES
Affiliation:
Department of Mathematics, University of Colorado, Boulder, CO 80309-0395, USA email Keith.Kearnes@Colorado.EDU
*
✉ghorvath@science.unideb.hu
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Abstract

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We prove that ${ \mathbb{Z} }_{{p}^{n} } $ and ${ \mathbb{Z} }_{p} [t] / ({t}^{n} )$ are polynomially equivalent if and only if $n\leq 2$ or ${p}^{n} = 8$. For the proof, employing Bernoulli numbers, we explicitly provide the polynomials which compute the carry-on part for the addition and multiplication in base $p$. As a corollary, we characterize finite rings of ${p}^{2} $ elements up to polynomial equivalence.

Keywords

polynomial equivalencefinite ringsBernoulli numberscarry for base p additioncarry for base p multiplication

MSC classification

Secondary: 13M10: Polynomials 13B25: Polynomials over commutative rings 11B68: Bernoulli and Euler numbers and polynomials
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 96 , Issue 2 , April 2014 , pp. 244 - 257
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Bulatov, A. A., ‘Polynomial clones containing the Mal’tsev operation of the groups ${ \mathbb{Z} }_{{p}^{2} } $ and ${ \mathbb{Z} }_{p} \times { \mathbb{Z} }_{p} $’, Mult.-Valued Log. 8 (2) (2002), 193–221.CrossRefGoogle Scholar
Clausen, T., ‘Theorem’, Astron. Nachr. 17 (1840), 351–352.Google Scholar
Fine, B., ‘Classification of finite rings of order ${p}^{2} $’, Math. Mag. 66 (4) (1993), 248–252.Google Scholar
Frisch, S., ‘Polynomial functions on finite commutative rings’, in: Advances in Commutative Ring Theory. Proc. 3rd Int. Conf., Fez, Morocco, Lecture Notes in Pure and Applied Mathematics, 205 (ed. Dobbs, D. E.) (Marcel Dekker, New York, 1999), 323–336.Google Scholar
Hobby, D. and McKenzie, R., The Structure of Finite Algebras, Contemporary Mathematics 76 (American Mathematical Society, Providence, RI, 1988).Google Scholar
Idziak, P. M. and Słomczyńska, K., ‘Polynomially rich algebras’, J. Pure Appl. Algebra 156 (1) (2001), 33–68.Google Scholar
Ireland, K. and Rosen, M., A Classical Introduction to Modern Number Theory (Springer, New York, 1982).Google Scholar
Istinger, M. and Kaiser, H. K., ‘A characterization of polynomially complete algebras’, J. Algebra 56 (1) (1979), 103–110.Google Scholar
Maurer, W. D. and Rhodes, J. L., ‘A property of finite simple nonabelian groups’, Proc. Amer. Math. Soc. 16 (1965), 552–554.Google Scholar
Rédei, L. and Szele, T., ‘Algebraischzahlentheoretische Betrachtungen über Ringe. I’, Acta Math. 79 (1947), 291–320.Google Scholar
Schottenfels, I. M., ‘Two nonisomorphic simple groups of the same order 20,160’, Ann. of Math. (2) 1 (1–4) (1899–1900), 147–152.Google Scholar
von Staudt, Ch., ‘Beweis eines Lehrsatzes, die Bernoullischen Zahlen betreffend’, J. reine angew. Math. 21 (1840), 372–374.Google Scholar