Ring Of Integer-valued Polynomials Research Articles

The Skolem property in rings of integer-valued rational functions

Let D be a domain and let Int(D) and IntR(D) be the ring of integer-valued polynomials and the ring of integer-valued rational functions, respectively. Skolem proved that if I is a finitely-generated ideal of Int(Z) with all the value ideals of I not being proper, then I=Int(Z). This is known as the Skolem property, which does not hold in Z[x]. One obstruction to Int(D) having the Skolem property is the existence of unit-valued polynomials. This is no longer an obstruction when we consider the Skolem property on IntR(D). We determine that the Skolem property on IntR(D) is equivalent to the maximal spectrum being contained in the ultrafilter closure of the set of maximal pointed ideals. We generalize the Skolem property using star operations and determine an analogous equivalence under this generalized notion.

Topological aspects of the ideal theory in rings of integer-valued polynomials

Let D D be a Dedekind domain with finite residue fields. We provide topological insights into certain classes of ideals of I n t ( D ) Int(D) lying over a given maximal ideal m \mathfrak m of D D . We completely determine invertible/divisorial ideals in terms of topological properties of subsets of the m \mathfrak m -adic completion of D D . Moreover, these results are naturally extended to overrings of I n t ( D ) Int(D) . As an application we provide explicit constructions of divisorial ideals of I n t ( D ) Int(D) which are not finitely generated.

Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains

Rings of integer-valued polynomials are known to be atomic, non-factorial rings furnishing examples for both irreducible elements for which all powers factor uniquely (absolutely irreducibles) and irreducible elements where some power has a factorization different from the trivial one.In this paper, we study irreducible polynomials F∈Int(R) where R is a discrete valuation domain with finite residue field and show that it is possible to explicitly determine a number S∈N that reduces the absolute irreducibility of F to the unique factorization of FS.To this end, we establish a connection between the factors of powers of F and the kernel of a certain linear map that we associate to F. This connection yields a characterization of absolute irreducibility in terms of this so-called fixed divisor kernel. Given a non-trivial element v of this kernel, we explicitly construct non-trivial factorizations of Fk, provided that k≥L, where L depends on F as well as the choice of v. We further show that this bound cannot be improved in general. Additionally, we provide other (larger) lower bounds for k, one of which only depends on the valuation of the denominator of F and the size of the residue class field of R.

Stable Centres of Iwahori-Hecke Algebras of Type A

A celebrated result of Farahat and Higman constructs an algebra FH which “interpolates” the centres \(Z(\mathbb {Z}S_{n})\) of group algebras of the symmetric groups Sn. We extend these results from symmetric group algebras to type A Iwahori-Hecke algebras, Hn(q). In particular, we explain how to construct an algebra FHq “interpolating” the centres Z(Hn(q)). We prove that FHq is isomorphic to \(\mathcal {R}[q,q^{-1}] \otimes _{\mathbb {Z}} {\Lambda }\) (where \(\mathcal {R}\) is the ring of integer-valued polynomials, and Λ is the ring of symmetric functions). The isomorphism can be described as “evaluation at Jucys-Murphy elements”, leading to a proof of a conjecture of Francis and Wang. This yields character formulae for the Geck-Rouquier basis of Z(Hn(q)) when acting on Specht modules.

Irreducibility of integer-valued polynomials in several variables

Let \(\underline{S}\) be an arbitrary subset of \(R^n\) where R is a domain with the field of fractions \(\mathbb {K}\). Denote the ring of polynomials in n variables over \(\mathbb {K}\) by \(\mathbb {K}[\underline{x}] \). The ring of integer-valued polynomials over \(\underline{S}\), denoted by Int\((\underline{S},R)\), is defined as the set of the polynomials of \(\mathbb {K}[\underline{x}] \), which maps \(\underline{S}\) to R. In this article, we study the irreducibility of the polynomials of Int\((\underline{S},R)\) for the first time in the case when R is a Unique Factorization Domain. We also show that our results remain valid when R is a Dedekind domain or, sometimes, any domain.

Bhargava factorials and irreducibility of integer-valued polynomials

The ring of integer-valued polynomials over a given subset S of ℤ (or Int(S,ℤ)) is defined as the set of polynomials in ℚ[x] which maps S to ℤ. In factorization theory, it is crucial to check the irreducibility of a polynomial. In this article, we make Bhargava factorials our main tool to check the irreducibility of a given polynomial f∈Int(S,ℤ)). We also generalize our results to arbitrary subsets of a Dedekind domain.

A note on integer-valued skew polynomials

Given an integral domain [Formula: see text] with quotient field [Formula: see text], the study of the ring of integer-valued polynomials [Formula: see text] has attracted a lot of attention over the past decades. Recently, Werner has extended this study to the situation of skew polynomials. To be more precise, if [Formula: see text] is an automorphism of [Formula: see text], one may consider the set [Formula: see text], where [Formula: see text] is the skew polynomial ring and [Formula: see text] is a “suitable” evaluation of [Formula: see text] at [Formula: see text]. For example, he gave sufficient conditions for [Formula: see text] to be a ring and study some of its properties. In this paper, we extend the study to the situation of the skew polynomial ring [Formula: see text] with a suitable evaluation, where [Formula: see text] is a [Formula: see text]-derivation. Moreover we prove, for example, that if [Formula: see text] is of finite order and [Formula: see text] is a Dedekind domain with finite residue fields such that [Formula: see text] is a ring, then [Formula: see text] is non-Noetherian.

An Elementary Proof of the Two-Generator Property for the Ring of Integer-Valued Polynomials

The ring of integer-valued polynomials has the two-generator property, which means that every finitely generated ideal may be generated by two elements. As the known proofs of this fact are rather complicated using strong topological arguments, we propose here a constructive proof obtained by means of elementary tools. Along the way, we also obtain constructive proofs of two other well-known facts: the finitely generated ideals of are characterized by their ideals of values and is a two-dimensional Prüfer domain.

Chains of prime ideals in power series rings

An ideal I of a commutative ring D with identity is called an SFT ideal if there exist a finitely generated ideal J with J ⊆ I and a positive integer k such that a k ∈ J for each a ∈ I . We prove that for a non-SFT maximal ideal M of an integral domain D , ht ( M 〚 X 〛 / M D 〚 X 〛 ) ≥ 2 ℵ 1 if either (1) D is a 1-dimensional quasi-local domain (in particular D is a 1-dimensional nondiscrete valuation domain) or (2) M is the radical of a countably generated ideal. In other words, if one of the conditions (1) and (2) is satisfied, then there is a chain of prime ideals in D 〚 X 〛 with length at least 2 ℵ 1 such that each prime ideal in the chain lies between M D 〚 X 〛 and M 〚 X 〛 . As an application, assuming the continuum hypothesis we show that if D is either the ring of algebraic integers or the ring of integer-valued polynomials on Z , then dim ⁡ D 〚 X 〛 = ht M 〚 X 〛 = ht ( M 〚 X 〛 / M D 〚 X 〛 ) = 2 ℵ 1 for every maximal ideal M of D .

Irreducibility of integer-valued polynomials I

Let be an arbitrary subset of a unique factorization domain R and be the field of fractions of R. The ring of integer-valued polynomials over S is the set This article is an effort to study the irreducibility of integer-valued polynomials over arbitrary subsets of a unique factorization domain. We give a method to construct special kinds of sequences, which we call d-sequences. We then use these sequences to obtain a criteria for the irreducibility of the polynomials in In some special cases, we explicitly construct these sequences and use these sequences to check the irreducibility of some polynomials in At the end, we suggest a generalization of our results to an arbitrary subset of a Dedekind domain.

The Skolem closure as a semistar operation

Skolem properties are usually defined on rings of integer-valued polynomials, and are thought to be about finitely generated ideals. We contend that Skolem properties are more naturally stated in terms of ⋆-ideals. By this method, we characterize the classes of ideals on which Int(ℤ) has the Skolem and strong Skolem properties. We extend the definitions of Skolem properties to rings comprising rational functions. We further define semistar operations which generalize the Skolem closure in this broader context. Finally we give some conditions under which a ring has Skolem properties.

Note on integer-valued polynomials on a residually cofinite subset

Let D be an integral domain with quotient field K, E a subset of K and X an indeterminate over K. The ring $$\mathrm {Int}(E,D):=\{f\in K[X]\;:\; f(E)\subseteq D\}$$ , of integer-valued polynomials on E with respect to D, is known to be a D-algebra. Obviously, $$\mathrm {Int}(D,D)=\mathrm {Int}(D)$$ , is the classical ring of integer-valued polynomials over D. In this paper, we study the faithful flatness, the local freeness and the calculation of the Krull dimension of integer-valued polynomials on a residually cofinite subset over locally essential domains. In particular, we prove that $$\mathrm {Int}(E,D)$$ is faithfully flat as a D-module and is of Krull dimension less than that of D[X], when D is a locally essential domain and $$E\subseteq D$$ is residually cofinite with D. Also, we get stronger results in the case of domains that are either almost Krull or t-almost Dedekind.

Integer-valued skew polynomials

For a commutative integral domain [Formula: see text] with field of fractions [Formula: see text], the ring of integer-valued polynomials on [Formula: see text] is [Formula: see text]. In this paper, we extend this construction to skew polynomial rings. Given an automorphism [Formula: see text] of [Formula: see text], the skew polynomial ring [Formula: see text] consists of polynomials with coefficients from [Formula: see text], and with multiplication given by [Formula: see text] for all [Formula: see text]. We define [Formula: see text], which is the set of integer-valued skew polynomials on [Formula: see text]. When [Formula: see text] is not the identity, [Formula: see text] is noncommutative and evaluation behaves differently than it does for ordinary polynomials. Nevertheless, we are able to prove that [Formula: see text] has a ring structure in many cases. We show how to produce elements of [Formula: see text] and investigate its properties regarding localization and Noetherian conditions. Particular attention is paid to the case where [Formula: see text] is a discrete valuation ring with finite residue field.

A graph-theoretic criterion for absolute irreducibility of integer-valued polynomials with square-free denominator

An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one (essentially different) factorization into irreducibles. In the case of the ring of integer-valued polynomials on a principal ideal domain D with quotient field K, we give an easy to verify graph-theoretic sufficient condition for an element to be absolutely irreducible and show a partial converse: the condition is necessary and sufficient for polynomials with square-free denominator.

Factorization theory in commutative monoids

This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.

Non-absolutely irreducible elements in the ring of integer-valued polynomials

Let R be a commutative ring with identity. An element is said to be absolutely irreducible in R if for all natural numbers n > 1, rn has essentially only one factorization namely If is irreducible in R but for some n > 1, rn has other factorizations distinct from then r is called non-absolutely irreducible. In this paper, we construct non-absolutely irreducible elements in the ring of integer-valued polynomials. We also give generalizations of these constructions.

A Survey on Fixed Divisors

In this article, we compile the work done by various mathematicians on the topic of the fixed divisor of a polynomial. This article explains most of the results concisely and is intended to be an exhaustive survey. We present the results on fixed divisors in various algebraic settings as well as the applications of fixed divisors to various algebraic and number theoretic problems. The work is presented in an orderly fashion so as to start from the simplest case of $\Z,$ progressively leading up to the case of Dedekind domains. We also ask a few open questions according to their context, which may give impetus to the reader to work further in this direction. We describe various bounds for fixed divisors as well as the connection of fixed divisors with different notions in the ring of integer-valued polynomials. Finally, we suggest how the generalization of the ring of integer-valued polynomials in the case of the ring of $n \times n$ matrices over $\Z$ (or Dedekind domain) could lead to the generalization of fixed divisors in that setting.

Integer-valued polynomials, Prüfer domains and the stacked bases property

To study the question of whether every two-dimensional Prüfer domain possesses the stacked bases property, we consider the particular case of the Prüfer domains formed by integer-valued polynomials. The description of the spectrum of the rings of integer-valued polynomials on a subset of a rank-one valuation domain enables us to prove that they all possess the stacked bases property. We also consider integer-valued polynomials on rings of integers of number fields and in this case we reduce the study of the stacked bases property to questions concerning 2×2-matrices.

Sets of lengths of factorizations of integer-valued polynomials on Dedekind domains with finite residue fields

Let D be a Dedekind domain with infinitely many maximal ideals, all of finite index, and K its quotient field. Let Int(D)={f∈K[x]|f(D)⊆D} be the ring of integer-valued polynomials on D.Given any finite multiset {k1,…,kn} of integers greater than 1, we construct a polynomial in Int(D) which has exactly n essentially different factorizations into irreducibles in Int(D), the lengths of these factorizations being k1, …, kn. We also show that there is no transfer homomorphism from the multiplicative monoid of Int(D) to a block monoid.

Integer-valued polynomials over block matrix algebras

In this paper, we state a generalization of the ring of integer-valued polynomials over upper triangular matrix rings. The set of integer-valued polynomials over some block matrix rings is studied. In fact, we consider the set of integer-valued polynomials [Formula: see text] for each [Formula: see text], where [Formula: see text] is an integral domain with quotient field [Formula: see text] and [Formula: see text] is a block matrix ring between upper triangular matrix ring [Formula: see text] and full matrix ring [Formula: see text]. In fact, we have [Formula: see text]. It is known that the sets of integer-valued polynomials [Formula: see text] and [Formula: see text] are rings. We state some relations between the rings [Formula: see text] and the partitions of [Formula: see text]. Then, we show that the set [Formula: see text] is a ring for each [Formula: see text]. Further, it is proved that if the ring [Formula: see text] is not Noetherian then the ring [Formula: see text] is not Noetherian, too. Finally, some properties and relations are stated between the rings [Formula: see text], [Formula: see text] and [Formula: see text].