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Bhargava Rings Over Subsets

  • I. Al-Rasasi
  • L. IzelgueEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 228)

Abstract

Let D be an integral domain with quotient field K and let E be any nonempty subset of K. The Bhargava ring over E at \(x\in D\) is defined by \(\mathbb {B}_x(E,D):=\{f\in K[X]\mid f(xX+e)\in D[X], \ \forall e\in E\}\). This ring is a subring of the ring of integer-valued polynomials over E. This paper studies \(\mathbb {B}_x(E,D)\) for an arbitrary domain D. we provide information about its localizations and transfer properties, describe its prime ideal structure, and calculate its Krull and valuative dimensions.

Keywords

Integer-valued polynomial Bhargava ring Prime ideal Localization Residue field Krull dimension Valuative dimension 

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranKingdom of Saudi Arabia
  2. 2.Department of Mathematics, Faculty of Sciences SemlaliaCadi Ayyad UniversityMarrakechMorocco

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