A note on 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.

Skolem properties, value-functions, and divisorial ideals

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 \(\mathbb{Q}\).

The study of factorization in integral domains has played an important role in commutative algebra for many years. Much of this work has con­centrated on the study of unique factorization domains (UFDs). Beyond the realm of UFDs, there is a large class of integral domains for which each nonunit can be factored as a product of irreducibles, yet the factorization may not be unique. Classically, Dedekind domains are such domains. A Dedekind domain D is a UFD if and only if its ideal class group is trivial. Hence the size of the class group becomes a measure of how far the domain D is from having “unique factorization”. The concept of the class group and its relation to factorization properties extends also to the more general class of Krull domains. It is within the context of these domains that we consider in this paper general questions concerning the lengths of factorization in the non-UFD setting.

We determine the relationship between the cardinality of a Noetherian integral domain and the cardinality of a residue field. One consequence of the main result is that it is provable in Zermelo–Fraenkel Set Theory with Choice (ZFC) that there is a Noetherian domain of cardinality ℵ1 with a finite residue field, but the statement “There is a Noetherian domain of cardinality ℵ2 with a finite residue field” is equivalent to the negation of the Continuum Hypothesis.

Let $D$ be an integral domain with quotient field $K$ and $\Omega$ a finite subset of $D$. McQuillan proved that the ring ${\rm Int}(\Omega,D)$ of polynomials in $K[X]$ which are integer-valued over $\Omega$, that is, $f\in K[X]$ such that $f(\Omega)\subset D$, is a Pr\"ufer domain if and only if $D$ is Pr\"ufer. Under the further assumption that $D$ is integrally closed, we generalize his result by considering a finite set $S$ of a $D$-algebra $A$ which is finitely generated and torsion-free as a $D$-module, and the ring ${\rm Int}_K(S,A)$ of integer-valued polynomials over $S$, that is, polynomials over $K$ whose image over $S$ is contained in $A$. We show that the integral closure of ${\rm Int}_K(S,A)$ is equal to the contraction to $K[X]$ of ${\rm Int}(\Omega_S,D_F)$, for some finite subset $\Omega_S$ of integral elements over $D$ contained in an algebraic closure $\bar K$ of $K$, where $D_F$ is the integral closure of $D$ in $F=K(\Omega_S)$. Moreover, the integral closure of ${\rm Int}_K(S,A)$ is Pr\"ufer if and only if $D$ is Pr\"ufer. The result is obtained by means of the study of pullbacks of the form $D[X]+p(X)K[X]$, where $p(X)$ is a monic non-constant polynomial over $D$: we prove that the integral closure of such a pullback is equal to the ring of polynomials over $K$ which are integral-valued over the set of roots $\Omega_p$ of $p(X)$ in $\bar K$.

An integral domain R is said to be weakly factorial if every nonunit of R is a product of primary elements.We give several conditions equivalent to R being weakly factorial.For example, we show that the following conditions are equivalent: (1) R is weakly factorial; (2) every convex directed subgroup of the group of divisibility of R is a cardinal summand; (3) if P is a prime ideal of R minimal over a proper principal ideal (x), then P has height one and (x)p n R is principal; (4) R = C\Rp , where the intersection runs over the height-one primes of R , is locally finite, and the i-class group of R is trivial.Throughout this note, R will be a commutative integral domain with identity having quotient field K. Suppose that R is a UFD.If S is a saturated multiplicatively closed subset of R, then S = {Xpx -pn\X a unit of R, n > 0, each (p) Y} for some subset Y X{X), where X{X) is the set of height-one prime ideals of R .Conversely, every such subset Y X{X) determines a saturated multiplicatively closed subset of R. If we set T = {Xpx -p"\X a unit of R , n > 0, each (p() G X(1) -Y}, then every nonzero element r of ii may be written uniquely up to units in the form r = st where s e S and t T. Stated in terms of the group of divisibility G(R) of R, (S) is a cardinal summand of G(R).In fact, (S) c (T) -G(R).(See the next paragraph for definitions.)Here G(R) is a cardinal sum of copies of Z, one for each height-one prime, and (S) is the sum of the cyclic summands corresponding to the primes from Y .The purpose of this paper is to characterize the integral domains with this property.We show that an integral domain R has the property that every convex directed subgroup of G(R) is a cardinal summand of G(R) if and only if every nonunit of R is a product of primary elements, that is, R is weakly factorial.Let R be an integral domain with quotient field K.The group of divisibility of R is the group G(R) = K*/U(R), where K* is the multiplicative group of K and U(R) is the group units of R. G(R) is partially ordered

It is proved that every prime factor algebra of the coordinate ring (?q(Mn(k)) of quantum n x n matrices over a field k is an integral domain (albeit not necessarily commutative) when q is not a root of unity.The same conclusion follows for the quantum groups tfq(SL"(k)) and c?q(GL"(k)).The proof uses a (/-analog of Sigurdsson's theorem bounding the Goldie ranks of prime factors of differential operator rings; this (/-analog in turn is based on results from the authors' recent work on (/-skew polynomial rings.

Let (R,m) be a complete discrete valuation ring with the finite residue field R∕m= Fq. Given a monic polynomial P(t)∈R[t] whose reduction modulo m gives an irreducible polynomial P‾(t)∈Fq[t], we initiate an investigation of the distribution of coker(P(A)), where A∈Matn(R) is randomly chosen with respect to the Haar probability measure on the additive group Matn(R) of n×n R-matrices. In particular, we provide a generalization of two results of Friedman and Washington about these random matrices. We use some concrete combinatorial connections between Matn(R) and Matn(Fq) to translate our problems about a Haar-random matrix in Matn(R) into problems about a random matrix in Matn(Fq) with respect to the uniform distribution. Our results over Fq are about the distribution of the P‾-part of a random matrix A‾∈Matn(Fq) with respect to the uniform distribution, and one of them generalizes a result of Fulman. We heuristically relate our results to a celebrated conjecture of Cohen and Lenstra, which predicts that given an odd prime p, any finite abelian p-group (i.e., Zp-module) H occurs as the p-part of the class group of a random imaginary quadratic field extension of Q with a probability inversely proportional to |AutZ(H)|. We review three different heuristics for the conjecture of Cohen and Lenstra, and they are all related to special cases of our main conjecture, which we prove as our main theorems.

Poonen proved an analogue for Drinfeld modules of the Mordell-Weil theorem. We shall generalize his results to the case of t t -modules which have the height property. Moreover, we also prove a Mordell–Weil theorem for certain t t -modules which lack the height property. In particular, let A A be the polynomial ring in one variable θ \theta over the finite field F q \mathbb {F}_q with the quotient field K K . Let ( E , ϕ ) (E, \phi ) be an iterated extension of the tensor powers of the Carlitz module defined over K K . We show that the F q [ t ] \mathbb {F}_q[t] -module E ( K ) E(K) is isomorphic to the direct sum of its torsion module, which is finite, with a free F q [ t ] \mathbb {F}_q[t] -module of rank ℵ 0 \aleph _0 .

Almost slender rings and compact rings

Let A be a commutative integral domain with quotient field L, and let R be a subdomain of A with quotient field K. Assuming that L is a Galois extension of K, Nagata required the condition for R to be normal when A is called a Galois extension of R (see p. 31, M. Nagata, Local Rings (Wiley, New York, 1962)). However in this paper, A is considered in the case that R is not necessarily assumed to be normal. We introduce the notion of cyclic Galois extensions of integral domains and investigate several properties of such ring extensions. In particular, we completely determine the seminormalization [Formula: see text] of A in an overdomain B such that both A ⊆B are cyclic Galois extensions of R.

Let R be a commutative ring and S a multiplicatively closed subset of R having no zero divisors. The pair ⟨ R , S ⟩ \langle R,S\rangle is said to be stable if the ring of fractions of R, S − 1 R {S^{ - 1}}R , defined by S is a ring for which all finitely generated projective modules are free. For a stable pair ⟨ R , S ⟩ \langle R,S\rangle assume 2 is a unit in R and V is a free R-module of dimension ⩾ 3 \geqslant 3 . This paper examines the action of a group automorphism of G L ( V ) GL(V) (the general linear group) on the elementary matrices relative to a basis B of V. In the case that R is a local ring, a Euclidean domain, a connected semilocal ring or a Dedekind domain whose quotient field is a finite extension of the rationals, we obtain a description of the action of the automorphism on all elements of G L ( V ) GL(V) .

The complexity of linear problems in fields

In this paper, we study orbits and fixed points of polynomials in a general skew polynomial ring [Formula: see text]. We extend results of the first author and Vishkautsan on polynomial dynamics in [Formula: see text]. In particular, we show that if [Formula: see text] and [Formula: see text] satisfy [Formula: see text], then [Formula: see text] for every formal power of [Formula: see text]. More generally, we give a sufficient condition for a point [Formula: see text] to be [Formula: see text]-periodic with respect to a polynomial [Formula: see text]. Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy.