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Automorphisms of Ideals of Polynomial Rings

  • Tiago Macedo
  • Thiago Castilho de Mello
Article
  • 126 Downloads

Abstract

Let R be a commutative integral domain with unit, f be a nonconstant monic polynomial in R[t], and \(I_f \subset R[t]\) be the ideal generated by f. In this paper we study the group of R-algebra automorphisms of the R-algebra without unit \(I_f\). We show that, if f has only one root (possibly with multiplicity), then \({{\mathrm{Aut}}}(I_f) \cong R^\times \). We also show that, under certain mild hypothesis, if f has at least two different roots in the algebraic closure of the quotient field of R, then \({{\mathrm{Aut}}}(I_f)\) is a cyclic group and its order can be completely determined by analyzing the roots of f.

Keywords

Automorphisms Commutative Algebra Ideals 

Mathematics Subject Classification

Primary 08A35 13A15 16W20 

Notes

Acknowledgements

The first author would like to thank CNPq and the second author would like to thank CNPq and Fapesp for their financial supports. The first author would also like to thank D. Nakano for bringing up the problem that motivated this paper, and both authors would like to thank J. Hahn and E. Wofsey for their important suggestions (Hahn and Wofsey 2015).

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Copyright information

© Sociedade Brasileira de Matemática 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  2. 2.Department of Science and TechnologyFederal University of São PauloSão José dos CamposBrazil

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