Commutative Rings and Integer-Valued Polynomials

First Joint International Meeting of AMS and UMI Pisa, ITALY Thursday, 13 June 2002 Program

9:00: David Dobbs:

On the strong going-between and going-down properties of ring homomorphisms 9:30: Bruce Olberding:

Reflexive Prüfer Domains in a Noetherian Setting 10:00: Moshe Roitman:

Complete integral closure and strongly divisorial prime ideals 10:20–10:45: BREAK 10:45: Marco Fontana:

Nagata rings, Kronecker function rings and related semistar operations 11:15: Scott Chapman:

Inside factorial monoids and integral domains 11:45: Martine Picavet-L’Hermitte:

Cale Bases in Algebraic Orders 12:15: Sylvia Wiegand:

Examples of integral domains inside power series rings 12:45: Mi Hee Park:

Power series rings over Strong Mori domains 1:05–3:00: LUNCH 3:00: Sarah Glaz:

Homological behavior of Gaussian rings with zero-divisor controlling conditions 3:30: Sébastien Pellerin:

On the overrings of half-factorial domains 4:00: Marco D’Anna:

Semigroups of plane singularities 4:20–4:45: BREAK 4:45: William W. Smith:

Factorization Problems in Int(E, D) 5:15: Sophie Frisch:

Integer-valued polynomials on matrices 5:45: Jean-Luc Chabert:

On the representation of integers as linear combinations of values of polynomials 6:15: Francesca Tartarone:

Strong Mori and Noetherian properties of Integer Valued Polynomial Rings 6:45: Paul-Jean Cahen:

Regular bases for B-Rings

On the strong going-between and going-down properties of ring homomorphisms

David E. Dobbs and Gabriel Picavet

Throughout, A is a fixed (commutative integral) domain. If a domain B contains A, then f_{B} denotes the inclusion map A ↪ B. We consider analogies between the logically independent properties of strong going-between (SGB) and going-down (GD). THEOREM 1. The following four conditions are equivalent (for the given domain A): (1)f_{A[u]} is SGB for each element u in the quotient field of A; (2)f_{B} is SGB for each valuation overring B of A; (3)f_{B} is SGB for each domain B containing A; (4)A is a going-down domain. REMARK 2. (a) If B is a flat overring of A, then f_{B} is SGB. (b) If a domain B that contains A is A-flat, then f_{B} need not be (S)GB. THEOREM 3. Let B be a domain that contains A. Then: (a) If f_{B} is universally SGB, then f_{B} is universally GD. (b) If B is an overring of A and f_{B} is universally GD, then f_{B} is universally SGB. THEOREM 4. If A is locally finite-dimensional, then f_{B} is universally SGB for each domain B that contains A and is algebraic over A if and only if A is a universally going-down domain. LEMMA 5. If A is a one-dimensional S-domain and X, Y are algebraically independent indeterminates over A, then A[X] ↪ A[X, Y] is not SGB. THEOREM 6. If R is a nonzero ring and X is an indeterminate over R, then R ↪ R[X] is not universally SGB. Also, transfer results for the (universally) SGB property are obtained relative to pullbacks and Nagata ring constructions.

Reflexive Prüfer Domains in a Noetherian Setting

Bruce Olberding

If R is a commutative ring, an R-module M is reflexive if the canonical homomorphism ϕ : M → Hom_{R}(Hom_{R}(M, R), R) is an isomorphism. If R is an integral domain and ϕ is an isomorphism for all submodules M of finitely generated free R-modules, then R is a reflexive domain. Reflexive domains were studied extensively in the sixties and seventies by a number of researchers, including H. Bass and E. Matlis. Recently the class of reflexive domains has been shown to figure prominently in the classification of integral domains which support Warfield duality, a duality between certain categories of torsion-free modules that has its origins in abelian group theory. In this talk, we show how Warfield duality and reflexivity arise naturally in Prüfer overrings of complete local Noetherian domains. We also indicate how this leads to constructions of “large” FGC rings, those rings R for which every finitely generated module can be written as a direct sum of cyclic modules. Here, “large” means R has infinitely many maximal ideals and Krull dimension greater than one.

Complete integral closure and strongly divisorial prime ideals

Valentina Barucci, Stefania Gabelli and Moshe Roitman

It is well known that a domain without proper strongly divisorial ideals is completely integrally closed. In this talk we show that a domain without prime strongly divisorial ideals is not necessarily completely integrally closed, although this property holds under some additional assumptions.

Inside factorial monoids and integral domains

Scott T. Chapman, Franz Halter-Koch and Ulrich Krause

We investigate two classes of monoids and integral domains, called inside and outside factorial, whose definitions are closely related in a divisor-theoretic manner to the concept of unique factorization. We prove that a monoid is outside factorial if and only if it is a Krull monoid with torsion class group, and that it is inside factorial if and only if its rootclosure is a rational generalized Krull monoid with torsion class group. We determine the structure of Cale bases of inside factorial monoids and characterize inside factorial monoids among weakly Krull monoids.

These characterizations carry over to integral domains. Inside factorial orders in algebraic number fields are characterized by several other factorization properties.

Nagata Rings, Kronecker Function Rings and Related Semistar Operations

Marco Fontana

In 1994 Okabe and Matsuda generalize the concept of star operation by introducing the more “flexible” notion of semistar operation. After that paper new developments of the multiplicative theory of ideals have been realized and successfully applied to analyze the structure of different classes of integral domains (cf. for instance Okabe-Matsuda (1997), Matsuda-Sugatani (1995), Fontana-Huckaba (2000), Fontana-Loper (2001 and 2002), and Halter-Koch (2001)). Semistar operations of a special type appear naturally in relation with the general construction of Kronecker function rings and Nagata function rings. More precisely, given a semistar operation ⋆ on an integral domain D with quotient field K, the Kronecker function ring Kr(D, ⋆) (⊆ K(X)) [respectively, the Nagata function ring Na(D, ⋆) (⊆ K(X)] induces naturally a “distinguished” semistar operation ⋆_{a} [respectivally, ˜⋆] on D such that F Kr(D, ⋆) ∩ K = F^{⋆_{a}} [respectivally, F Na(D, ⋆) ∩ K = F ˜⋆], for each finitely generated fractional ideal F of D. One of the purposes of this talk is to study these semistar operations, to investigate the interplay of Kr(D, ⋆) and ⋆_{a} with Na(D, ⋆) and ˜⋆ and to show a “parallel” behavior of these pairs of rings. The equality of Nagata function ring and Kronecker function ring characterizes, in the classical Noetherian case, the Dedekind domains. The equality of Nagata function ring Na(D, ⋆) with Kronecker function ring Kr(D, ⋆) characterizes the Prüfer ⋆-multiplication domains.

Refences

Fontana, M., Huckaba, J., Localizing systems and semistar operations, in “Non-Noetherian Commutative Ring Theory” (S. Chapman and S. Glaz, Eds.), Kluwer Academic Publishers 2000, Chapter 8. Fontana, M., Jara, P., Santos, E., Prüfer ⋆-multiplication domains and semistar operations, Submitted. Fontana, M. Loper, K. A., Kronecker function rings: a general approach, in “Ideal Theoretic Methods in Commutative Algebra” (D.D. Anderson, I.J. Papick, Eds.), M. Dekker Lecture Notes Pure Appl. Math. 220 (2001), 189–206. Fontana, M., Loper, K.A., Nagata rings, Kronecker function rings and related semistar operations, Submitted.

Garcia, J.M., Jara, P., Santos, E., Prüfer ⋆-multiplication domains and torsion theories, Comm. Algebra 27 (1999), 1275–1295. Halter-Koch, F., Localizing systems, module systems and semistar operations, J. Algebra 238 (2001), 723–761

Cale Bases in Algebraic Orders

Martine Picavet-L’Hermitte

According to S.T. Chapman, F. Halter-Koch and U. Krause, a subset Q of a monoid M is a Cale basis for M if, for every nonunit x ∈ M, some power of x has a unique factorization into elements of Q. Such a monoid is called inside factorial.

Let R be a non-maximal order in a finite algebraic number field. Let \overline{R} be the integral closure of R and (R : \overline{R}) be its conductor. We denote by J the intersection of all maximal ideals P of R such that there is more than one maximal ideal of \overline{R} lying above P. Then the set M of all elements of R comaximal with J is an inside factorial monoid. We exhibit a Cale basis Q whose elements are irreducible and primary and a positive integer N such that x^{N} has a unique factorization into elements of Q for each x ∈ R \ \{0\}. When Spec(\overline{R}) → Spec(R) is bijective, R \ {0} is an inside factorial monoid.

In the quadratic case, we get either N = es or N = 2es, where e is the exponent of the class group of R and s is the order of U(\overline{R})/U(R), depending of the form of (R : \overline{R}).

Examples of integral domains inside power series rings

William Heinzer, Christel Rotthaus and Sylvia Wiegand

We present examples of Noetherian and non-Noetherian integral domains which can be built inside power series rings. Given a power series ring R^{∗} over a Noetherian integral domain R and given a subfield L of the total quotient ring of R^{∗} with R ⊆ L, we construct subrings A and B of L such that B is a localization of a nested union of polynomial rings over R and B ⊆ A := L ∩ R^{∗}. We show in certain cases that flatness of a related map on polynomial rings is equivalent to the Noetherian property for B. Moreover if B is Noetherian, then B = A. We use this construction to obtain explicit examples of both Noetherian and non-Noetherian integral domains.

Power series rings over Strong Mori domains

Mi Hee Park

Let R be an integral domain with quotient field K and let F (R) be the set of nonzero fractional ideals of R. Then on F (R) the w-operation is defined by A ↦ A_{w} = {x ∈ K|Jx ⊆ A for some finitely generated ideal J with J_{v} = R}. A fractional ideal A is called a w-ideal if A = A_{w}. An integral domain R is called a Strong Mori domain if it satisfies the ascending chain condition on integral w-ideals.

In this paper we show that (1) if R is a Noetherian domain, then R[[{X_{λ}}_{λ∈Λ}]]_{1} is a Strong Mori domain, (2) if R is a Strong Mori domain, then R[[{X_{λ}}_{λ∈Λ}]]_{1} is a Mori domain for any arbitrary set Λ, but (3) there exists a Strong Mori domain R such that R[[X]] is not a Strong Mori domain.

Homological behavior of Gaussian rings with zero-divisor controlling conditions

Sarah Glaz

Let R be a commutative ring, and let f = f(x) be a polynomial in one variable over R. Denote by c(f), the content of f, the ideal of R generated by the coefficients of f. R is called a Gaussian ring if c(fg) = c(f)c(g) for any two polynomials f and g over R. Gaussian rings were defined by Tsang in 1965, who also proved that an integral domain is Gaussian iff it is a Prufer domain. A number of other authors, among them Gilmer, Dan Anderson and Camillo investigated the ring theoretic behavior of Gaussian rings. This talk revolves around the speaker’s recent results concerning the homological behavior of Gaussian rings. In particular, coherent Gaussian rings are characterized in terms of their weak global dimension and their small finitistic dimension. The talk also explores the impact of several zero-divisor controlling conditions on the homological properties of Gaussian rings.

On the overrings of half-factorial domains

Nathalie Gonzalez and Sébastien Pellerin

When R is a half-factorial domain, it is well-known that the polynomial extension R[X] is not necessarily a half-factorial domain. Yet, the question is still widely opened for the case of overrings. Namely, if R is a half-factorial domain and if S is an overring of R (that is, a domain containing R with the same quotient field), then is S a half-factorial domain? Despite the answer is false in general, we aim to know which overrings are half-factorial.

The aim of this talk is to study the behavior of Coykendall’s boundary map on a halffactorial domain and, then, to derive conditions for these overrings to be half-factorial. We pay special attention to the case of extensions R ⊂ S such that: for each nonzero nonunit s ∈ S, there is a unit u of S with us ∈ R. Finally, we apply our results for polynomial extensions of type A + XB[X] ⊂ B[X].

Semigroups of plane singularities

Marco D’Anna

This is a joint work in progress with V. Barucci and R. Forberg. Let k be an algebraically closed field of characteristic zero and let R = k[[X_{1}, …, X_{n}]]/I be a one dimensional reduced ring. Since the integral closure \overline{R} of R in its quotient field is of the form \overline{R} ≅ k[[t_{1}]] × ⋯ × k[[t_{d}]] (where d is the number of minimal primes of R), it is possible to associate to R a value semigroup v(R) ⊆ ℕ^{d}.

The study of this kind of rings is connected to the study of curve singularities and the use of the value semigroup is a classical method to get information on the ring.

In the case of plane singularities (i.e. R = k[[X, Y]]/(f)) it is possible to find a strict connection between the value semigroup and the multiplicity tree of the ring. In a previous paper of the authors, is studied the case f irreducible and it is given a criterion to check if a semigroup is the value semigroup of a plane singularity and, conversely, to construct such semigroup starting from the multiplicity sequence of the ring.

Factorization Problems in Int(E,D)

William W. Smith

Let D be an integral domain with quotient field K. For a non-empty subset E of D, set

R = Int(E, D) = {f(x) ∈ K[x] | f(E) ⊆ D}.

Various problems and results will be presented concerning factorization of particular elements of R. Particular attention will be focused on the case where E is a finite set.

Integer-valued polynomials on matrices (preliminary report)

Sophie Frisch

If D is an integral domain with quotient field K, we consider polynomials in K[x] that map every n × n matrix with entries in D to a matrix with entries in D, and similarly polynomials in K⟨x_{1}, …, x_{m}⟩ (in several non-commuting indeterminates) that result in a matrix with entries in D whenever matrices with entries in D are plugged in. In the case of one variable, this leads to a study of the null ideal (or characteristic ideal) of a matrix over a finite commutative ring; in the case of several variables, the theory of polynomial identity rings in non-zero characteristic can be used to construct non-trivial examples of smallest possible degree. In either case, the ring of such matrix-integer-valued polynomials is trivial if and only if the corresponding usual ring of integer-valued polynomials is trivial.

On the representation of integers as linear combinations of values of polynomials

Jacques Boulanger and Jean-Luc Chabert

Erdös proved that every integer N may be written as a linear combination with coefficients ±1 of the r first squares (r depends on N). Recently, Bodoni, Duchet and Lefranc showed that in this assertion one may replace the squares by the cubes or by the nth powers for any n ; they also considered integer-valued polynomials.

Here we extend Erdös result to algebraic integers in the following way. Let K be a number field and let A be the corresponding ring of integers. Let f ∈ K[X] be any polynomial of degree n such that f(Z) ⊆ A. We study the set of the integers N of K that may be written as linear combinations of the r first values of f with units of K as coefficients, that is:

N = ∑_{j=1}^{r} ε_{j}f(j) with ε_{j} ∈ U(K).

For example, if K = Q, Q[i] or Q[\sqrt{−3}], we show that all integers of K may have such a representation as soon as the ideal of A generated by the values of f on Z is the domain A itself. For instance, f = X^{n} or f = \frac{X(X−1)···(X−n+1)}{n!}.

Strong Mori and Noetherian properties of Integer Valued Polynomial Rings

Mi Hee Park and Francesca Tartarone

Let D be a domain with quotient field K. An ideal J of D is called a Glaz-Vasconcelos ideal (GV-ideal) if J is finitely generated and J_{v} = D (where J_{v} = (D: (D: J)) is the divisorial closure of J). The w-closure of I is the ideal

I_{w} := {x ∈ K; xJ ⊆ I for some J is a GV ideal of D}.

The w-operation defined by I ↦ I_{w}, is a ∗-operation of finite character over the set of D-fractional ideals.

A domain D is Strong Mori if it satisfies the ascending chain condition (a.c.c.) on wideals. Thus, Strong Mori domains are a natural generalization of Noetherian domains and also form a subclass of Mori domains (which satisfy the a.c.c. on integral divisorial ideals). We consider the ring Int(D) := {f ∈ K[X]; f(D) ⊆ D} of integer-valued polynomials over D.

Establishing whether Int(D) is Noetherian is an open question. We look for conditions on D in order that Int(D) is Strong Mori. We show, in some cases, the equivalence between the Noetherian and the Strong Mori properties for Int(D). For instance, when D is local and Int(D) ≠ D[X] or when D is Noetherian it occurs that Int(D) is Strong Mori if and only if it is Noetherian. In the case that (D: D′) ≠ (0) (where D′ is the integral closure of D), we give a complete characterization of the domains D such that Int(D) is Strong Mori.

More generally, we reduce the problem of Int(D) being Strong Mori to the problem of Int(D_{2}) being Noetherian, where D_{2} is a suitable overring of D.

Regular bases for B-Rings

Paul-Jean Cahen and Julie Yeramian

Let D be a domain with quotient field K. For each x ∈ D, we consider the “B-ring” 𝔹_{x}(D) formed by the polynomials f ∈ K[X] such that f(xX +a) belongs to D[X] for each a ∈ D. The ring of integer-valued polynomials Int(D) is the union of all such B-rings. We describe regular bases for those rings over a discrete valuation domain D. It follows that, over a Dedekind domain, B-rings are Noetherian, whereas, in general, Int(D) is not. For instance 𝔹_{2}(ℤ) = ℤ[X][\frac{X(X−1)}{2}.