Non-triviality conditions for integer-valued polynomial rings on algebras

Peruginelli, Giulio; Werner, Nicholas J.

doi:10.1007/s00605-016-0951-8

Cited by 10 publications

(3 citation statements)

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“…In order to describe the family of Dedekind domains lying between Z[X] and Q[X], we briefly recall the notion of integer-valued polynomials on algebras (see [8,23], for example). Let D be an integral domain with quotient field K and A a torsion-free D algebra.…”

Section: Polynomial Dedekind Domainsmentioning

confidence: 99%

Metrizability of spaces of valuation domains associated to pseudo-convergent sequences

Peruginelli

Spirito

2021

J. Algebra Appl.

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Let [Formula: see text] be a valuation domain of rank one with quotient field [Formula: see text]. We study the set of extensions of [Formula: see text] to the field of rational functions [Formula: see text] induced by pseudo-convergent sequences of [Formula: see text] from a topological point of view, endowing this set either with the Zariski or with the constructible topology. In particular, we consider the two subspaces induced by sequences with a prescribed breadth or with a prescribed pseudo-limit. We give some necessary conditions for the Zariski space to be metrizable (under the constructible topology) in terms of the value group and the residue field of [Formula: see text].

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Section: Polynomial Dedekind Domainsmentioning

confidence: 99%

Metrizability of spaces of valuation domains associated to pseudo-convergent sequences

Peruginelli

Spirito

2021

J. Algebra Appl.

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“…In pursuit of this problem, we will deal with more general rings of integer-valued polynomials, and describe when these rings are trivial, in the sense that they are equal to ordinary rings of polynomials. We refer to the papers [25,28] for studies on related problems. Definition 1.4.…”

Section: Note That Intmentioning

confidence: 99%

Integral Closure of Rings of Integer-Valued Polynomials on Algebras

Peruginelli

Werner

2014

Commutative Algebra

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Let D be an integrally closed domain with quotient field K. Let A be a torsion-free D-algebra that is finitely generated as a D-module. For every a in A we consider its minimal polynomial µa(X) ∈ D[X], i.e. the monic polynomial of least degree such that µa(a) = 0. The ring IntK (A) consists of polynomials in K[X] that send elements of A back to A under evaluation. If D has finite residue rings, we show that the integral closure of IntK(A) is the ring of polynomials in K[X] which map the roots in an algebraic closure of K of all the µa(X), a ∈ A, into elements that are integral over D. The result is obtained by identifying A with a D-subalgebra of the matrix algebra Mn(K) for some n and then considering polynomials which map a matrix to a matrix integral over D. We also obtain information about polynomially dense subsets of these rings of polynomials.

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“…In the theory of polynomial mappings on commutative rings, there are two notable subtopics, namely, polynomial functions on finite rings, and rings of integervalued polynomials. Here, we are concerned with generalizations of these two topics to polynomial mappings on non-commutative rings, as proposed by Loper and Werner [8], and developed further by Werner [11][12][13][14], Peruginelli [9,10], and the present author [3][4][5], among others. More particularly, we will investigate connections between null ideals of polynomials on finite non-commutative rings and integer-valued polynomials on non-commutative rings.…”

Section: Introductionmentioning

confidence: 99%