Integer-valued polynomials on commutative rings and modules

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.

Irreducible polynomials and full elasticity in rings of integer-valued polynomials

In this paper the following question is studied: when can a homogeneous element of a Grassmann algebra be represented as the sum of two decomposable elements? For an exterior algebra over a field necessary and sufficient conditions of such a representation are obtained, over an arbitrary integral domain several necessary conditions, and over Krull rings also several sufficient conditions. In particular, it is established that the only rings such that the verification of 2-decomposability is carried out in the same way as over fields are the fields, that is, there are no “2-Plucker” rings.

We introduce two new notions of “ P P -ordering” and use them to define a three-parameter generalization of the usual factorial function. We then apply these notions of P P -orderings and factorials to some classical problems in two distinct areas, namely: 1) the study of integer-valued polynomials and 2) P P -adic analysis. Specifically, we first use these notions of P P -orderings and factorials to construct explicit Pólya-style regular bases for two natural families of rings of integer-valued polynomials defined on an arbitrary subset of a Dedekind domain. Second, we classify “smooth” functions on an arbitrary compact subset S S of a local field, by constructing explicit interpolation series (i.e., orthonormal bases) for the Banach space of functions on S S satisfying any desired conditions of continuous differentiability or local analyticity. Our constructions thus extend Mahler’s Theorem (classifying the functions that are continuous on Z p \mathbb {Z}_p ) to a very general setting. In particular, our constructions prove that, for any ϵ > 0 \epsilon >0 , the functions in any of the above Banach spaces can be ϵ \epsilon -approximated by polynomials (with respect to their respective Banach norms). Thus we obtain the non-Archimedean analogues of the classical polynomial approximation theorems in real and complex analysis proven by Weierstrass, de la Vallée-Poussin, and Bernstein. Our proofs are effective.

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 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.

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 result of Farahat and Higman shows that there is a “universal” algebra, FH , interpolating the centres of symmetric group algebras, Z ( ℤ S n ) . We explain that this algebra is isomorphic to ℛ ⊗ Λ , where ℛ is the ring of integer-valued polynomials and Λ is the ring of symmetric functions. Moreover, the isomorphism is via “evaluation at Jucys–Murphy elements”, which leads to character formulae for symmetric groups. Then, we generalise this result to wreath products Γ ≀ S n of a fixed finite group Γ . This involves constructing wreath-product versions ℛ Γ and Λ ( Γ * ) of ℛ and Λ , respectively, which are interesting in their own right (for example, both are Hopf algebras). We show that the universal algebra for wreath products, FH Γ , is isomorphic to ℛ Γ ⊗ Λ ( Γ * ) and use this to compute the p -blocks of wreath products.

Generating ideals in rings of integer-valued polynomials

Let D be a Dedekind domain with quotient field K. The ring of integer-valued polynomials on D is the subring Int(D) = { f ∈ K[X]: f(D) ⊆ D} of the polynomial ring K[X]. The Polya-Ostrowski group PO(D) of D is a subgroup of the class group of D generated by the well-known factorial ideals n! D of D. A regular basis of Int(D) is a D-module basis consisting of one polynomial of each degree. It is well known that Int(D) has a regular basis if and only if the group PO(D) is trivial, if and only if the D-module Int n (D) = { f ∈ Int(D): degf ≤ n} is free for all n. In this paper we provide evidence for and prove special cases of the conjecture that, if PO(D) is finite, then the natural density of the set of nonnegative integers n such that Int n (D) is free exists, is rational, and is at least 1∕ | PO(D) | . Moreover, we compute this density or determine a conjectural value for several examples of Galois number fields of degrees 2, 3, 4, 5, and 6 over \(\mathbb{Q}\).

Another Prüfer Ring of Integer-Valued Polynomials

Let [Formula: see text] be a principal ideal domain with infinite spectrum such that for every nonzero prime ideal [Formula: see text] of [Formula: see text], the residue field [Formula: see text] is finite. Let [Formula: see text] be the quotient field of [Formula: see text]. We investigate sets of lengths in the ring of integer-valued polynomials on [Formula: see text], [Formula: see text]M, D[Formula: see text]. For every multiset of integers [Formula: see text], we explicitly construct an element of Int([Formula: see text]) with exactly [Formula: see text] essentially different factorizations into irreducible elements of Int([Formula: see text] whose lengths are [Formula: see text]. Furthermore, we show that Int([Formula: see text]) is not a transfer Krull domain. These results spark off the study of sets of lengths in the rings Int([Formula: see text]) ≠Int([Formula: see text]), where [Formula: see text] is an infinite subset of [Formula: see text].

A submonoid S of a given monoid H is called monadic if it is a divisor-closed submonoid of H generated by one element (i.e., there is some (non-zero) b ∈ H such that S is the smallest divisor-closed submonoid of H such that b ∈ S). In this paper we study monoids and domains whose monadic submonoids are Krull monoids. These monoids resp. domains are called monadically Krull. Every Krull monoid is a monadically Krull monoid, but the converse is not true. We provide several types of counterexamples and present a few characterizations for monadically Krull monoids. Furthermore, we show that rings of integer-valued polynomials over factorial domains are monadically Krull. Finally, we investigate the connections between monadically Krull monoids and generalizations of SP-domains.

Several different generalizations of finite factorization domains (i.e., integral domains where every nonzero nonunit has only finitely many divisors up to associates) have been defined for commutative rings with zero divisors. We study these notions in the context of commutative monoid rings with zero divisors, utilizing semigroup theory to simultaneously generalize and extend many past results about “finite factorization” properties in commutative polynomial rings. Along the way, we expand upon the general theory of factorization in commutative rings with zero divisors, providing new characterizations and results about several kinds of “finite factorization rings.”

Torsion theories over commutative rings