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Automorphisms of Ideals of Polynomial Rings

Automorphisms of Ideals of Polynomial Rings Let R be a commutative integral domain with unit, f be a nonconstant monic polynomial in R[t], and $$I_f \subset R[t]$$ I f ⊂ 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 $$I_f$$ I f . We show that, if f has only one root (possibly with multiplicity), then $${{\mathrm{Aut}}}(I_f) \cong R^\times $$ Aut ( I f ) ≅ R × . 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 $${{\mathrm{Aut}}}(I_f)$$ Aut ( I f ) is a cyclic group and its order can be completely determined by analyzing the roots of f. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Brazilian Mathematical Society, New Series Springer Journals

Automorphisms of Ideals of Polynomial Rings

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References (12)

Publisher
Springer Journals
Copyright
Copyright © 2017 by Sociedade Brasileira de Matemática
Subject
Mathematics; Mathematics, general; Theoretical, Mathematical and Computational Physics
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/s00574-017-0046-8
Publisher site
See Article on Publisher Site

Abstract

Let R be a commutative integral domain with unit, f be a nonconstant monic polynomial in R[t], and $$I_f \subset R[t]$$ I f ⊂ 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 $$I_f$$ I f . We show that, if f has only one root (possibly with multiplicity), then $${{\mathrm{Aut}}}(I_f) \cong R^\times $$ Aut ( I f ) ≅ R × . 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 $${{\mathrm{Aut}}}(I_f)$$ Aut ( I f ) is a cyclic group and its order can be completely determined by analyzing the roots of f.

Journal

Bulletin of the Brazilian Mathematical Society, New Series – Springer Journals

Published: Jul 4, 2017

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