On a property of the ideals of the polynomial ring R[x]

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Abstract

Let R be a commutative ring with unity 1 ≠ 0. In this paper we introduce the definition of the first derivative property on the ideals of the polynomial ring R[x]. In particular, when R is a finite local ring with principal maximal ideal m ≠ {0} of index of nilpotency e, where 1 < e ≤ |R/m| + 1, we show that the null ideal consisting of polynomials inducing the zero function on R satisfies this property. As an application, when R is a finite local ring with null ideal satisfying this property, we prove that the stabilizer group of R in the group of polynomial permutations on the ring R[x]/(x 2), is isomorphic to a certain factor group of the null ideal.

Original languageEnglish
Pages (from-to)1-12
Number of pages12
JournalInternational Electronic Journal of Algebra
Volume31
Issue number31
DOIs
Publication statusPublished - 17 Jan 2022

Keywords

  • Commutative rings
  • dual numbers
  • finite local ring
  • finite permutation group
  • Henselian ring
  • null ideal
  • null polynomial
  • permutation polynomial
  • polynomial permutation
  • polynomial ring

ASJC Scopus subject areas

  • Algebra and Number Theory

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