Open Problems in Commutative Ring Theory

Abstract

This chapter consists of a collection of open problems in commutative algebra. The collection covers a wide range of topics from both Noetherian and non-Noetherian ring theory and exhibits a variety of research approaches, including the use of homological algebra, ring theoretic methods, and star and semistar operation techniques. The problems were contributed by the authors and editors of this volume, as well as other researchers in the area.

1 Introduction

This chapter consists of a collection of open problems in commutative algebra. The collection covers a wide range of topics from both Noetherian and non-Noetherian ring theory and exhibits a variety of research approaches, including the use of homological algebra, ring theoretic methods, and star and semistar operation techniques. The problems were sent to us, in response to our request, by the authors of the articles in this volume and several other researchers in the area. We also included our own contributions. Some of these problems appear in other chapters of this volume, while others are unrelated to any of them, but were considered important by their proposers. The problems were gathered by the contributors from a variety of sources and include long-standing conjectures as well as questions that were generated by the most recent research in the field. Definitions and clarifying comments were added to make the problems more self-contained, but, as a rule, if unidentified notions are used, the reader can find the relevant definitions in the cited references. The purpose of this chapter is to generate research, and we hope that the problems proposed here will keep researchers happily busy for many years.

The underlying assumption is that all rings are commutative with 1 (and 1 ≠ 0), all modules are unital, all groups are abelian, and the term “local ring” refers to a not-necessarily Noetherian ring with a unique maximal ideal. Several notions and ring constructions appear in a number of the proposed problems. For the reader’s convenience, we mention a few definitions and sources of information here:

Let R be a commutative ring and let G be an abelian group written multiplicatively. The group ring RG is the free R module on the elements of G with multiplication induced by G. An element x in RG has a unique expression $x =\sum _{g\epsilon G}x_{g}g,$ where x _g ε R and all but finitely many x _g are zero. Addition and multiplication in RG are defined analogously to the standard polynomial operations. Basic information about commutative group rings may be found in [60, 84] and [63, Chap. 8 (Sect.2)].

Let D be a domain with quotient field K; Int(D) denotes the ring of integer-valued polynomials, that is, $\mathrm{Int}(D) =\{ f \in K[X]\mid f(D) \subseteq D\}.$ More generally, for a subset E of K, $\mathrm{Int}(E,D) =\{ f \in K[X]\mid f(E) \subseteq D\}$, and thus, $\mathrm{Int}(D) = \mathrm{Int}(D,D)$. For several indeterminates, $\mathrm{Int}(D^{n}) =\{ f \in K[X_{1},\ldots,X_{n}]\mid f(D^{n}) \subseteq D\}.$ For a D-algebra A, containing D, $\mathrm{Int}_{K}(A) =\{ f \in K[X]\mid f(A) \subseteq A\}.$ Note that $\mathrm{Int}_{K}(A)$ is contained in Int(D) if and only if A ∩ K = D. Basic information about integer-valued polynomials may be found in [19].

Basic information on the star and semistar operations that appear in some of these problems may be found in [59, Sects. 32 and 34] and [95].

Basic information on the integral closure of ideals in Noetherian rings that is used in some of these problems may be found in [81].

Finally, for interested readers, we provide the list of contributors in the Acknowledgment section at the end of this chapter.

2 Open Problems

Problem 1.

Glaz [67] and Bazzoni and Glaz [14] consider, among other properties, the finitistic and weak global dimensions of rings satisfying various Prüfer conditions. The Prüfer conditions under considerations are:

• (1) R is semihereditary (i.e., finitely generated ideals of R are projective).

• (2) w.dim R ≤ 1. 

• (3) R is arithmetical (i.e., ideals of R _m are totally ordered by inclusion for all maximal ideals m of R). 

• (4) R is Gaussian (i.e., c(fg) = c(f)c(g) for all f, g ε R[x]).

• (5) R is locally Prüfer (i.e., R _p is Prüfer for every prime ideal p of R).

• (6) R is Prüfer (i.e., every finitely generated regular ideal of R is invertible).

Let mod R be the set of all R-modules admitting a projective resolution consisting of finitely generated projective modules. The finitistic projective dimension of R is defined as fp.dim R = sup {proj.dim _R M: M εmod R and proj.dim _R M < ∞}. In general, fp.dim R ≤ w.dim R, and if R is a local coherent regular ring, then fp.dim R = w.dim R [67, Lemma 3.1]:

Problem 1a.

Let R be a Prüfer ring. Is fp.dim R ≤ 1?

The answer is affirmative for Gaussian rings [67, Theorems 3.2 and 14, Proposition 5.3]. It is also clearly true for Prüfer domains.

Problem 1b.

Let R be a total ring of quotients. Is fp.dim R = 0?

Note that a total ring of quotients is always a Prüfer ring, so this question asks if for this particular kind of Prüfer ring, fp.dim R is not only at most equal to 1, but is actually equal to 0. This is true for a local Gaussian total ring of quotients [67, Theorem 3.2 (Case 1)]. More information and a detailed bibliography on the subject may be found in [66, 68].

Problem 2.

Using the results obtained for the finitistic projective dimension of rings (see previous problem) with various Prüfer conditions, it is possible to determine the values of the weak global dimensions of rings under certain Prüfer conditions [14, 67], but many questions are not yet answered:

Problem 2a.

If R is a Gaussian ring, then is w.dim R = 0, 1, or ∞?

This is the case for coherent Gaussian rings [67, Theorem 3.3] (and actually, more generally, for coherent Prüfer rings [14, Proposition 6.1]), arithmetical rings [97 and 14, remark in the last paragraph], and a particular case of Gaussian rings [14, Theorem 6.4].

Problem 2b.

If R is a total ring of quotients, is w.dim R = 0, 1, or ∞?

This is true if R is coherent [68, Corollary 6.7], in which case w.dim R = 0 or ∞, and also holds for one example of a non-coherent ring [68, Example 6.8].

In general:

Problem 2c.

What are the values of w.dim R when R is a Prüfer ring?

Additional information and references on the subject may be found in [68].

Problem 3.

Is the integral closure of a one-dimensional coherent domain in its field of quotients a Prüfer domain?

This question, which also appears in [23, Problem 65], was posed by Vasconcelos. It had been answered positively in many, but not all, cases. A useful reference that will lead to many other useful references is [63, Chap. 5 (Sect. 3) and Chap. 7 (Sect. 4)].

Problem 4.

A ring R is a finite conductor ring if aR ∩ bR and (0: c) are finitely generated ideals of R for all elements a, b, and c in R. A ring R is aquasi-coherent ring if a ₁ R ∩ … ∩ a _n R and (0: c) are finitely generated ideals of R for all elements a ₁, …, a _n and c in R. Examples of both classes of rings include all coherent rings, UFDs, GCD domains, G-GCD domains (i.e., domains in which the intersection of two invertible ideals is an invertible ideal), and the still more general G-GCD rings (i.e., rings in which principal ideals are projective and the intersection of two finitely generated flat ideals is a finitely generated flat ideal). For more information on these classes of rings see references [64, 65]. Let G be a multiplicative abelian group and let RG be the group ring of G over R. In the group ring setting where R is a domain, characterizations of group rings as UFDs and GCD domains were obtained in [61, Theorems 6.1, 6.4, and 7.17]. In the case where R is a ring with zero divisors, however, the behavior of the finite conductor and quasi-coherent properties has been only partially described. Specifically, in the general ring setting, both properties descend from RG to R [65, Proposition 3.2], and the question of ascent from R to RG reduces to the situation where G is finitely generated [65, Proposition 3.1]. This, however, does not solve the problem of ascent for either property. Even in the case where R is a G-GCD ring and G is an infinite cyclic group, ascent is unknown.

Problem 4a.

Assume that R is a G-GCD ring and G is a finitely generated abelian group. Do the finite conductor and the quasi-coherent properties ascend from R to RG?

Further explorations of these conditions in the group ring setting may shed light on a more general problem:

Problem 4b.

Are the finite conductor and quasi-coherent properties for rings distinct?

Other useful references include [51, 69].

Problem 5.

Let R be a commutative ring and let G be an abelian group with the property that the order of every element of G is invertible in R. Then w.dim RG = w.dim R + rank G [33, Theorem] and [62, Theorem 2]. With the aid of this formula, it is possible to characterize von Neumann regular and semihereditary group rings RG [62, Corollaries 1 and 2].

Problem 5a.

Is there a similar formula that relates the global dimension of RG to the global dimension of R in combination with some invariant of the group G?

In this direction, there is one classical result, Maschke’s theorem [84, page 47], which pertains to a special case of semisimple rings (i.e., rings of global dimension zero): Let G be a finite group and let k be a field. Then k G is a semisimple ring if and only if the characteristic of k does not divide the order of the group G. Beyond this, characterizations of group rings of finite global dimension are not known, even for the simple cases of global dimension zero (semisimple rings) or one (hereditary rings).

Problem 5b.

Find characterizations of semisimple and hereditary group rings.

Other useful references include [69, 103].

Problem 6.

The generalization of the notion of a Cohen–Macaulay and a Gorenstein ring from the Noetherian to the non-Noetherian context is a recent development. As such, the questions of when a group ring RG, where R is a commutative ring and G is an abelian group, is Cohen–Macaulay or Gorenstein have yet to be investigated. The articles [72, 80] introduce the notions of non-Noetherian Cohen–Macaulay and non-Noetherian Gorenstein, respectively. An excellent survey of these theories which discusses the underlying homological framework is found in [79].

Problem 7.

This problem arises within the context of work done by Hamilton and Marley [72] to characterize non-Noetherian Cohen–Macaulay rings, as well as work of Hummel and Marley [80] to characterize non-Noetherian Gorenstein rings. In light of the role that homological dimensions, infinite finite free resolutions of modules, and local (co)homology played in the development of these theories for local coherent rings, Hummel [79] posed the question:

Is there a non-Noetherian characterization of local complete intersection rings such that (coherent) local Gorenstein rings are complete intersections and such that (coherent) local complete intersection rings are Cohen–Macaulay?

André [10] provided a characterization of non-Noetherian complete intersection rings parallel to its following Noetherian counterpart using André–Quillen homology: Let (R, m)be a local Noetherian ring with residue field k, and let β _i  = dim _k Tor _i ^R ( k, k) be the i th Betti number. R is a complete intersection if and only if the Betti numbers appear in the following equality of power series: $\sum \beta _{i}x^{i} = \frac{(1+x)^{r}} {(1-x^{2})^{s}}$ with $0\ < r - s = \mathrm{dim}\ R$.

While able to prove a similar characterization for non-Noetherian rings, André showed there was no relation between the integers r and s above in the non-Noetherian case.

In [79] one can find more background on this problem, as well as potential definitions or characterizations of non-Noetherian complete intersection rings.

Problem 8.

Let (R, M) be a Noetherian local ring. R is said to be quasi-complete if for any decreasing sequence $\{A_{n}\}_{n=1}^{\infty }$ of ideals of R and each natural number k, there exists a natural number s _k with $A_{s_{k}} \subseteq (\mathop{\bigcap }\nolimits _{n=1}^{\infty }A_{n}) + M^{k}$. If this condition holds for any decreasing sequence $\{A_{n}\}_{n=1}^{\infty }$ of ideals of R with $\mathop{\bigcap }\nolimits _{n=1}^{\infty }A_{n} = 0$, then R is called weakly quasi-complete (in which case, we actually have $A_{s_{k}} \subseteq M^{k}$). Now, if R is complete, then R is quasi-complete, which implies that R is weakly quasi-complete. Also R is quasi-complete if and only if each homomorphic image of R is weakly quasi-complete. The implication “R complete implies that R is quasi-complete” was first proved by Chevalley [24, Lemma 7]; for a proof in this volume see [4, Theorem 1.3]. Note that a DVR is quasi-complete, but need not be complete. More generally, a one-dimensional Noetherian local domain is (weakly) quasi-complete if and only if it is analytically irreducible [4, Corollary 2.2].

Problem 8a.

Is a weakly quasi-complete ring quasi-complete?

Problem 8b.

Let k be a field and $R = k[X_{1},\ldots,X_{n}]_{(X_{1},\ldots,X_{n})},n \geq 2.$ Is R (weakly) quasi-complete?

Regarding Problem 8b, in [4, Conjecture 1] it is conjectured that R is not weakly quasi-complete and in [4, Example 2.1] this is shown to be the case if k is countable. Note that R not being quasi-complete is equivalent to the existence of a height-one prime ideal P of $k[[X_{1},\ldots,X_{n}]]$ with $P \cap k[X_{1},\ldots,X_{n}] = 0.$ For additional information see [4].

Problem 9.

A commutative ring R is said to be a McCoy ring if each finitely generated ideal I ⊆ Z(R) (where Z(R) denotes the set of zero divisors of R) has a nonzero annihilator. In 1980, Akiba proved that if R is an integrally closed reduced McCoy ring, then the polynomial ring R[X] is also integrally closed [2, Theorem 3.2]. He also proved that if R _M is an integrally closed domain for each maximal ideal M, then R[X] is integrally closed [2, Corollary 1.3]. In addition he provided an example of a reduced ring R that is not a McCoy ring but locally is an integrally closed domain [2, Example]. Combining his results, one has that if R is a reduced ring such that R _M is an integrally closed McCoy ring for each maximal ideal M, then R[X] is integrally closed [78, page 103].

Does there exist an integrally closed reduced ring R such that R _M is an integrally closed McCoy ring for each maximal ideal M, but R is not a McCoy ring and it is not locally a domain?

Problem 10.

With the notation and definitions of the previous problem, a ring R has (A _n ) if each ideal I ⊆ Z(R) that can be generated by n (or fewer) elements has a nonzero annihilator. Example 2.5 in [90] shows that for each n ≥ 2, there are reduced rings which have (A _n ) but not (A _n+1). An alternate restriction on zero divisors is (a.c.): R has (a.c.) if for each pair of elements r, s ∈ R, there is an element t ∈ R such that Ann(r, s) = Ann(t). The rings in Examples 2.2 and 2.4 of [90] show that there are reduced McCoy rings that do not have (a.c.), and reduced rings with (a.c.) that are not McCoy.

Do there exist reduced rings that have both (a.c.) and (A _n ) for some n ≥ 2 that are not McCoy rings?

Problem 11.

Let P be a nonzero prime ideal of a Prüfer domain R. Then, it is known that PS is a divisorial prime in each overring R ⊆ S ⊆ R _P if and only if PR _P is principal and each P-primary ideal (of R) is a divisorial ideal of R.

Problem 11a.

Characterize when P (and R) is such that P ∩ T is a divisorial prime of T for each Prüfer domain T ⊆ R with the same quotient field as R.

Note that no assumption has been made about PS being, or not being, divisorial for $R \subsetneq S \subseteq R_{P}$. This is true for each P if $\mathbb{Z} \subseteq R \subseteq \mathbb{Q}$.

Problem 11b.

Characterize when P (and R) is such that there is no Prüfer domain T ⊆ R with the same quotient field as R such that P ∩ T is a divisorial prime of T.

More information about the problem may be found in [45].

Problem 12.

Let D be an almost Dedekind domain with a non-invertible maximal ideal M. Then D(X) is also an almost Dedekind domain and MD(X) is a non-invertible maximal ideal with corresponding residue field F(X) where $F = D/M$. Let R be the pullback of F[X]_(X) over MD(X). Then R is a Prüfer domain, $R \subsetneq D(X) \subsetneq D(X)_{MD(X)}$, MD(X) is a divisorial prime ideal of R, it is not a divisorial (prime) ideal of D(X), but MD(X) _MD(X) is a divisorial prime ideal of D(X) _MD(X). In contrast, if P is a nonzero non-maximal prime of the ring of entire functions E and N is a maximal ideal that contains P, then $E \subsetneq E_{N} \subsetneq E_{P}$, P( = P ²) is not a divisorial ideal of E, PE _N is a divisorial prime ideal of E _N , and PE _P is not a divisorial (prime) ideal of E _P .

Problem 12a.

Does there exist a Prüfer domain R with a nonzero prime P such that there is a countably infinite chain of overrings $R = R_{0} \subsetneq R_{1} \subsetneq R_{2} \subsetneq \cdots \subsetneq R_{P}$ where for all n ≥ 0, PR _2n is a divisorial ideal of R _2n and PR _2n+1 is an ideal of R _2n+1 that is not divisorial? If such a chain exists, is PS divisorial or not divisorial as an ideal of $S =\bigcup R_{m}(\subseteq R_{P})$?

Problem 12b.

Does there exist a Prüfer domain R with a nonzero prime P such that there is a countably infinite chain of underrings $R = R_{0} \supsetneq R_{1} \supsetneq R_{2} \supsetneq \cdots \supsetneq T =\bigcap R_{m}$ with T having the same quotient field as R where for all n ≥ 0, $P \cap R_{2n}$ is a divisorial ideal of R _2n and P ∩ R _2n+1 is not a divisorial ideal of R _2n+1? If such a chain exists, is P ∩ T divisorial or not divisorial as an ideal of T?

More information about the problem may be found in [45].

Problem 13.

A domain D with field of quotients K is called a straight domain if for every overring S of D, S∕PS is torsion-free over D∕P, for every prime ideal P of D.

Problem 13a.

If R _P and R∕P are straight domains, does this imply that R + PR _P is a straight domain?

A domain D is called divided if PD _P  = P for every prime ideal P of D; D is called locally divided if D _M is divided for every maximal ideal M of D. An answer to Question 13a may shed light on an open question posed in [30]:

Problem 13b.

Does there exist a straight domain which is not locally divided?

Extending these definitions, analogously, to general rings, we note that there exists a straight ring which is not a domain and not locally divided. For more details, see [32, Sect. 2].

Problem 14.

Comparing the Krull dimensions of the ring Int(D) of integer-valued polynomials and of the classical ring D[X] of polynomials with coefficients in D, it is known that dim$(\mathrm{Int}(D)) \geq \dim (D[X]) - 1$, with possibility of equality [19, Example V.1.12]. The question of an upper bound remains open: It is conjectured to be equal to dim(D[X]). 

Problem 15.

Considering a one-dimensional local Noetherian domain D with finite residue field, when does Int(D) satisfy the almost strong Skolem property?

See [19, Chap. VII] for a survey of Skolem properties. It is known that it is enough that D be analytically irreducible and was recently shown that D must be unibranched [20] leaving open the question of a necessary and sufficient condition.

Problem 16.

A sequence $\{a_{n}\}_{n\geq 0}$ of integers is said to be self (simultaneously) ordered if for all positive integers $m,n:\prod _{ k=0}^{n-1}(a_{n} - a_{k})$ divides $\prod _{k=0}^{n-1}(a_{m} - a_{k}).$ If f is a nonconstant polynomial of $\mathbb{Z}[X],$ distinct from ± X, then, for every $x \in \mathbb{Z}$, the sequence $\{f^{{\ast}n}(x)\}_{n\geq 0}$ (where f ^∗n denotes the nth iterate of f) is self-ordered [1, Proposition 18]. For instance, the sequence {q ⁿ} _n ≥ 0, with q ≠ 0, ±1, obtained with f(X) = qX and x = 1, is self-ordered. Aside from the (infinitely many) sequences obtained with such a dynamical construction are the three following “natural” sequences:$\left \{(-1)^{n}\left [\frac{n} {2} \right ]\right \}_{n\geq 1}\;,\;\{n^{2}\}_{ n\geq 0}\;,$ and $\left \{\frac{n(n+1)} {2} \right \}_{n\geq 0}$. Also, if the sequence {a _n } _n ≥ 0 is self-ordered, then so is $\{ba_{n} + c\}_{n\geq 0}$, for all integers $b\not =0$ and c. We note the importance of the word “natural” (although not defined), for instance, the “natural candidates” $\{n^{2}\}_{n\geq 1}$ and $\{n^{k}\}_{n\geq 0}\text{ for }k \geq 3$ are not self-ordered. On the other hand, one can construct infinitely many “artificial” self-ordered sequences by the following ad hoc construction: choose two distinct integers a ₀ and a ₁ and, for n ≥ 1, define inductively a _n+1 to be any integer, distinct from the a _k ’s for 0 ≤ k ≤ n, such that $\prod _{k=0}^{n-1}(a_{n} - a_{k})$ divides $a_{n+1} - a_{0}.$ These integers can be chosen to be prime numbers, thanks to Dirichlet’s theorem, and hence, there are infinitely many self-ordered sequences contained in the set $\mathbb{P}$ of prime numbers, although $\mathbb{P}$ itself cannot be self ordered. The question is:

Are there any other “natural” self-ordered sequences neither obtained by a dynamical construction nor by an affine map applied to the three previous examples?

More details can be found in [22, Sect. 5.2, Q.2].

Problem 17.

The nth Bhargava factorial [15] associated to an infinite subset E of $\mathbb{Z}$ is the integer n!  _E such that $\frac{1} {n!_{E}}$ is the generator of the fractional ideal formed by the leading coefficients of the polynomials $f(X) \in \mathrm{Int}(E, \mathbb{Z})$ with degree ≤ n [22, Sect. 3]. These factorials have the following properties:

• 1. 0!  _E  = 1.

• 2. $\forall n \geq 0$, n! divides n!  _E .

• 3. $\forall n,m \geq 0$, $n!_{E} \times m!_{E}$ divides (n + m)!  _E .

Mingarelli [92] called abstract factorials a sequence $\{n!_{a}\}_{n\geq 0}$ of positive integers satisfying these three properties. Clearly, such sequences are non-decreasing, but although there cannot be three consecutive equal terms [92, Lemma 8], one may have $k!_{a} = (k + 1)!_{a}$ for infinitely many k [92, Proposition 12].

Are there subsets E of $\mathbb{Z}$ such that the sequence of Bhargava’s factorials $\{n!_{E}\}_{n\geq 0}$ is not ultimately strictly increasing? [22, Sect. 4.3, Q.1].

Note that, for $E =\{ n^{3}\mid n \geq 0\}$, one has $3!_{E} = 4!_{E} = 504$, yet the sequence $\{n!_{E}\}_{n\geq 0}$ is ultimately strictly increasing.

Problem 18.

Given an abstract factorial $\{n!_{a}\}_{n\geq 0}$ as in the previous question, one may define a generalization of the constant e [92, Definition 17] by $e_{a} =\sum _{n\geq 0} \frac{1} {n!_{a}}.$ For Bhargava’s factorial associated to a subset E one denotes this number by e _E . For example, if $E = \mathbb{N}^{(2)} =\{ n^{2}\mid n \geq 0\}$, then $e_{\mathbb{N}^{(2)}} = \mathit{e}\mathrm{+\frac{1} {\mathit{e}}}$. The constant e _a is always irrational [92, Theorem 28].

For which infinite subset E is e _E a transcendental number [22, Sect. 6.5]?

Problem 19.

Int(D) is known to be a free D module if D is a Dedekind domain. TV domains, defined by Houston and Zafrullah in [77], are domains in which t-ideals coincide with v-ideals. TV PvMD domains were extensively studied in [82]. For a Krull domain, or more generally for a TV PvMD domain, Int(D) is known to be locally free and hence a flat D module [37].

Problem 19a.

Is Int(D) a flat D module for any domain D?

Problem 19b.

Is Int(D) a free D-module for any domain D?

Problem 20.

Let D be an integral domain. The canonical D-algebra homomorphism $\mathrm{Int}(D)^{\otimes _{D}n}\longrightarrow \mathrm{Int}(D^{n})$ is known to be an isomorphism if Int(D) is free or if D is locally free and $\mathrm{Int}(D_{\mathfrak{m}}) = \mathrm{Int}(D)_{\mathfrak{m}}$ for every maximal ideal $\mathfrak{m}$ of D [35].

Is this canonical morphism always injective? surjective?

Problem 21.

Let D be an integral domain. Does Int(D) always have a unique structure of a D-D-biring such that the inclusion $D[X]\longrightarrow \mathrm{Int}(D)$ is a homomorphism of D-D-birings?

This is the case if the canonical D-algebra homomorphism $\mathrm{Int}(D)^{\otimes _{D}n}\longrightarrow \mathrm{Int}(D^{n})$ is an isomorphism for all n [36].

Problem 22.

Let D ⊆ A be an extension of domains. Let B be the quotient field of A and $\mathrm{Int}(A) =\{ f \in B[X]\mid f(A) \subseteq A\}$ be the ring of integer-valued polynomials of A and similarly Int(A ⁿ) be the ring of integer-valued polynomials in several indeterminates. If $\mathrm{Int}(D^{n}) \subseteq \mathrm{Int}(A^{n})$ for all positive integer n, then the extension D ⊆ A is said to be almost polynomially complete [35].

If $\mathrm{Int}(D) \subseteq \mathrm{Int}(A)$, does it follow that the extension D ⊆ A is almost polynomially complete?

Problem 23.

Recall that, for a set S of nonnegative integers, the natural density δ(S) of S is defined to be $\ \delta (S) =\lim _{n\rightarrow \infty }\frac{\vert \{a\in S:a<n\}\vert } {n},$ provided the limit exists.

Let K be a number field and $\mathcal{O}_{K}$ be the corresponding ring of algebraic integers. Consider the natural density $\delta (\mathrm{Int}(\mathcal{O}_{K}))$ of the set of nonnegative integers n such that $\mathrm{Int}_{n}(\mathcal{O}_{K})$ is free ($\mathrm{Int}_{n}(\mathcal{O}_{K})$ denotes the $\mathcal{O}_{K}$-module formed by the integer-valued polynomials of degree at most n).

Prove or disprove the following conjecture [38]: $\delta (\mathrm{Int}(\mathcal{O}_{K}))$ exists, is rational, and is at least $1/\mathrm{Card}(\mathrm{PO}(\mathcal{O}_{K}))$ (where $\mathrm{PO}(\mathcal{O}_{K}))$ denotes the Pólya group of $\mathcal{O}_{K}$). Then compute $\delta (\mathrm{Int}(\mathcal{O}_{K})).$

Problem 24.

Consider the ring $\mathrm{Int}(E, \mathbb{Z})$ where E is the set formed by the elements of a sequence $\{u_{n}\}_{n\geq 0}$ of integers determined by a recursion $u_{n+1} = au_{n} + bu_{n-1}$ and initial values u ₀, u ₁. 

Problem 24a.

Compute a regular basis (i.e., a basis with nth term of degree n) of the $\mathbb{Z}$-module $\mathrm{Int}(E, \mathbb{Z})$.

Problem 24b.

Compute the characteristic sequence of $\mathrm{Int}(E, \mathbb{Z})$ with respect to each prime p, that is, the sequence $\{\alpha _{p}(n)\}_{n\geq 0}$, where α _p (n) is the p-adic valuation of the fractional ideal consisting of 0 and the leading coefficients of elements of $\mathrm{Int}(E, \mathbb{Z})$ of degree no more than n.

Problem 24c.

Determine the asymptotic behavior of this sequence, that is, compute the limit $\lim _{n\rightarrow \infty }\alpha (n)/n$ (the limit exists, by Fekete’s lemma, since the sequence {α(n)} is superadditive).

Based on the results of Coelho and Parry [25] answers are known if b = ±1, which includes the cases of the Fibonacci and Lucas numbers [83, 102]; however the method used there does not seem to extend to general a, b. This question can, of course, be extended to higher-order recursive sequences.

Problem 25.

There has been much recent progress in understanding $\mathrm{Int}_{\mathbb{Q}}(M_{n}(\mathbb{Z}))$ where $M_{n}(\mathbb{Z})$ is the ring of n × n matrices with integer coefficients, beginning with Frisch’s contribution [48] and more recently those of Peruginelli [98] and of Peruginelli and Werner [99]. These descriptions relate $\mathrm{Int}_{\mathbb{Q}}(M_{n}(\mathbb{Z}))$ to polynomials integer valued on algebraic integers or divisible by irreducibles modulo $d\mathbb{Z}[x]$. There remains however, as in the previous problem, to:

Problem 25a.

Compute a regular basis of $\mathrm{Int}_{\mathbb{Q}}(M_{n}(\mathbb{Z})).$

Problem 25b.

Compute its characteristic sequence with respect to each prime p. 

Problem 25c.

Determine the asymptotic behavior of this sequence.

Problem 26.

Let D be an integral domain and let A be a torsion-free D-algebra that is finitely generated as a D-module. We consider the ring $\mathrm{Int}_{K}(A)$ of polynomials, with coefficients in the quotient field K of D, that are integer-valued over A, that is, $\mathrm{Int}_{K}(A) =\{ f \in K[X]\mid f(A) \subseteq A\}$. For every a ∈ A, we denote by μ _a (X) ∈ D[X] the minimal polynomial of a over D. Each polynomial f in the pullback D[X] +μ _a (X)K[X] is obviously such that f(a) ∈ A. Hence, $\bigcap _{a\in A}(D[X] +\mu _{a}(X)K[X]) \subseteq \mathrm{Int}_{K}(A).$

For which algebras A as above does this inclusion become an equality?

Note that equality holds for the D-algebra A = M _n (D) of n × n matrices over D [99, Remark 3.4]. The ring $\mathrm{Int}_{K}\left (M_{n}(D)\right )$ has been studied in several places; see for instance [46, 98].

Problem 27.

Let D be an integral domain with quotient field K. Let A be a torsion-free D-algebra, containing D, finitely generated as a D-module, and such that K ∩ A = D. Letting B = K ⊗ _D A (i.e., the ring of fractions $\frac{a} {d}$ with a ∈ A and d ∈ D, d ≠ 0), one can consider the set $\mathrm{Int}(A) =\{ f \in B[X]\mid f(A) \subseteq A\}.$ Working with polynomials in B[X], one assumes that the indeterminate X commutes with all elements of B and that polynomials are evaluated with X on the right; see [85] for more details on polynomials with non-commuting coefficients. Note that $\mathrm{Int}_{K}(A) = \mathrm{Int}(A) \cap K[X].$ The set Int(A) is always a left A-module, but it is not clear whether it has a ring structure when A is not commutative. A sufficient condition for Int(A) to be a ring is that each element of A can be written as a finite sum $\sum _{i}c_{i}u_{i}$, where each u _i is a unit of A and each c _i is central in B [106]. Examples of algebras that meet this condition include the matrix rings M _n (D) and group rings DG where G is a finite group. Yet this condition is not necessary [107].

Problem 27a.

Give an example of a D-algebra A such that Int(A) is not a ring (possibly relaxing some of the conditions on A, for instance, that A is finitely generated as a D-module).

Problem 27b.

In [107], it is conjectured that Int(A) is always a ring when D has finite residue rings. Prove this conjecture or give a counterexample in this case.

Problem 28.

In [89], it is shown that if A is the ring of integers of an algebraic number field, then $\mathrm{Int}_{\mathbb{Q}}(A)$ is Prüfer. More generally, let D be an integral domain, let K be its quotient field, and let A be a D-algebra. If Int _K (A) is a Prüfer domain contained in Int(D), then Int(D) must also be Prüfer, and hence D must be integrally closed. This condition is not sufficient: Int _K (A) need not even be integrally closed and [99] gives some general theorems regarding the integral closure of Int _K (A).

Determine when this integral closure is Prüfer.

Note that Loper [87] determined such a criterion for the classical ring of integer-valued polynomials.

Problem 29.

In the early 1990s, J.D. Sally gave expository talks on the question of which rings lie between a Noetherian domain D and its quotient field F [74]. The manuscript [74] provides abundant evidence that when the dimension of D is greater than one, the class of rings between D and F is rich in interesting Noetherian and non-Noetherian rings. A narrower problem, which remains open, is the following:

Describe the integrally closed rings between a two-dimensional Noetherian domain and its quotient field.

Work on this problem is surveyed in [96]. The evidence suggests that this class of rings is quite complicated. A framework for describing the integrally closed rings between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$ is given in [88].

Problem 30.

Let R be a ring and let n be a positive integer. A proper ideal I of R is called an n -absorbing ideal if whenever the product $x_{1}x_{2}\cdots x_{n+1} \in I$ for $x_{1},x_{2},\ldots,x_{n+1} \in R$, then there are n of the x _i ’s whose product is in I. Clearly, a 1-absorbing ideal is just a prime ideal. A proper ideal I of R is called a strongly n-absorbing ideal if whenever $I_{1}I_{2}\cdots I_{n+1} \subseteq I$ for ideals $I_{1},I_{1},\ldots,I_{n+1}$ of R, then there are n of the ideals I _i ’s whose product is a subset of I. Obviously, a strongly n-absorbing ideal is an n-absorbing ideal.

Problem 30a.

Let I be an n-absorbing ideal of R. Is I a strongly n-absorbing ideal of R?

Note that if n = 1 the answer is obviously positive and if n = 2 a positive answer is contained in [11, Theorem 2.13].

Problem 30b.

Let I be an n-absorbing ideal of R. Is rad$(I)^{n} \subseteq I$?

Note that if I is an n-absorbing ideal, then so is rad(I). For n = 2, the positive answer to this question is proved in [11, Theorem 2.4].

Problem 30c.

Let I be an n-absorbing ideal of R. Is I[X] an n-absorbing ideal of the polynomial ring R[X]?

Note that if n = 2, then the answer is affirmative [7, Theorem 4.15].

Problem 31.

Let F∕K be a transcendental field extension, and let X be an indeterminate over F. By a result of Halter-Koch [70, Theorem 2.2], $A =\bigcap _{V }V (X) =\bigcap _{V }V [X]_{M_{V }[X]}$, (where V ranges over the valuation rings of F∕K and M _V is the maximal ideal of V ), is a Bézout domain. However, A is not the Kronecker function ring of a domain with quotient field F (and hence does not arise directly from an e.a.b. star operation). Fabbri and Heubo-Kwegna [39] show this issue can be circumvented when $F = K(X_{0},X_{1},\ldots,X_{n})$ is a purely transcendental extension of K by introducing the notion of projective star operations, which are glued together from traditional star operations on affine subsets of projective n-space. The ring A is then a “projective” Kronecker function ring of the projective analogue of the b-operation. More generally, e.a.b. projective star operations give rise to projective Kronecker function rings. The theory of projective star operations is worked out in [39] for projective n-space, but the following problem remains open:

Develop projective star operations for projective varieties (i.e., for the case where F∕K is a finitely generated field extension that need not be purely transcendental)

Problem 32.

Let D be a domain and let I be a nonzero ideal of D. Recall that I is called stable if I is invertible in its endomorphism ring E(I) = (I: I), and the integral domain D is called (finitely) stable if each nonzero (finitely generated) ideal is stable. As usual, the ideal I is called divisorial if $I = I^{v} = (D: (D: I)),$ and D is called a divisorial domain if each nonzero ideal is divisorial. A Mori domain is a domain satisfying the ascending chain condition on divisorial ideals. Clearly Noetherian and Krull domains are Mori. It has been proved in [54] that a stable domain is one-dimensional if and only if it is Mori. Since Mori domains satisfy the ascending chain condition on principal ideals, a Mori domain D is Archimedean, that is, $\bigcap _{n\geq 0}x^{n}D = (0)$, for each non-unit x ∈ D. The class of Archimedean domains includes also completely integrally closed domains and one-dimensional domains.

Problem 32a.

Is a stable Archimedean domain one-dimensional?

The answer is positive in the semilocal case [54], so that a semilocal stable Archimedean domain is Mori (see also [50, Theorem 2.17]). Hence, a way of approaching this problem is trying to see if for stable domains the Archimedean property localizes. In general the Archimedean property does not pass to localizations. For example, the ring of entire functions is an infinite-dimensional completely integrally closed (hence, Archimedean) Bézout domain which is not locally Archimedean, because an Archimedean valuation domain is one-dimensional.

If D is a Mori stable domain and I is a nonzero ideal of D, we have $I^{v} = (x,y)^{v}$, for some x, y ∈ I [50, Theorem 2.17]. Therefore, we can say that in a stable Mori domain, each divisorial ideal is 2-v-generated. Since a divisorial Mori domain is Noetherian, this result generalizes the fact that in a Noetherian domain that is stable and divisorial each ideal can be generated by two elements [50, Theorem 3.6].

Problem 32b.

Let D be a one-dimensional Mori domain such that each divisorial ideal is 2-v-generated. Is it true that each divisorial ideal is stable?

Note that the answer to this question is negative when D has dimension greater than one. For example, let D be a Krull domain. Then, each divisorial ideal of D is 2-v-generated [93, Proposition 1.2] and stability coincides with invertibility [50, Proposition 2.3]. Hence, each divisorial ideal of D is stable (i.e., invertible) if and only if D is locally factorial [17, Lemma 1.1].

Recall that a nonzero ideal I of an integral domain D is v -invertible if (I(D: I))^v = D, and I is called v -stable if I ^v is v-invertible as an ideal of E(I ^v), that is, $(I^{v}(E(I^{v}): I^{v}))^{v} = E(I^{v})$. Clearly, v-invertibility implies v-stability. If each nonzero ideal of D is v-stable, we say that D is v -stable [52]. Each nonzero ideal of a Krull domain is v-invertible; thus a Krull domain is v-stable. However, a Krull domain is stable if and only if it is a Dedekind domain, that is, it has dimension one. An example of a one-dimensional Mori domain that is v-stable but not stable is given in [53, Example 2.6].

Problem 32c.

Let D be a Mori domain such that each divisorial ideal is 2-v-generated. Is it true that D is v-stable?

Problem 33.

It is well known that if an integral domain D has finite character, then a locally invertible ideal is invertible. Conversely, if each locally invertible ideal is invertible, D need not have finite character (e.g., a Noetherian domain need not have finite character). However, a Prüfer domain such that each locally invertible ideal is invertible does have finite character. This fact was conjectured by Bazzoni [12, p 630] and proved by Holland et al. in [76]. (A simplified proof appears in [91]). Halter-Koch gave independently another proof in the more general context of ideal systems [71]. Other contributions were made by Zafrullah in [111] and by Finocchiaro et al. in [40]. Following Anderson and Zafrullah [6], an integral domain D is called an LPI domain if each locally principal nonzero ideal of D is invertible. Since a Prüfer domain is precisely a finitely stable integrally closed domain [50, Proposition 2.5], one is led to ask the following more general question (which appears in [13, Question 4.6]):

Assume that D is a finitely stable LPI domain. Is it true that D has finite character?

The answer to this question is positive if and only if the LPI property extends to fractional overrings [49, Corollary 15]. In particular, this holds when D is Mori or integrally closed. For an exhaustive discussion of this problem, see [49].

Problem 34.

Let D be an integral domain, let S be a multiplicatively closed set in D, and let D _S be the ring of fractions of D with respect to S. Consider $R = D + XD_{S}[X],$ the ring of polynomials over D _S in indeterminate X and with constant terms in D.

Recall that a GCD domain is an integral domain with the property that any two nonzero elements have a greatest common divisor and a PvMD is an integral domain with the property that every nonzero finitely generated ideal is t-invertible. It was shown in [26, Theorem 1.1] that R is a GCD domain if and only if D is a GCD domain and for all d ∈ D∖{0} GCD(d, X) exists. In [9, Theorem 2.5] it was shown that R is a Prüfer v-multiplication domain (for short, PvMD) if and only if D is a PvMD and the ideal (d, X) is t-invertible for all d ∈ D∖{0}. Next, recall that an integral domain is a v-domain if every nonzero finitely generated ideal is v-invertible. Note that if D is a v-domain, it is possible that D _S may not be a v-domain. For more information about v-domains consult [44].

Problem 34a.

Find necessary and sufficient conditions for $R = D + XD_{S}[X]$ to be a v-domain. Prove or disprove that R is a v-domain if and only if D is a v-domain and (d, X) is v-invertible for all d ∈ D∖{0}.

More generally:

Problem 34b.

Let A ⊆ B be an extension of integral domains, and let X be an indeterminate over B. Find necessary and sufficient conditions for the integral domain A + XB[X] to be a v-domain.

Problem 35.

An integral domain D is called an almost GCD (for short, AGCD) if for each pair x, y ∈ D∖{0}, there is an integer n = n(x, y) (depending on x and y) such that x ⁿ D ∩ y ⁿ D is principal. The theory of AGCD domains runs along lines similar to that of GCD domains; see [5] for more information and a list of references on the topic. An integral domain D is a domain of finite t -character if every nonzero non-unit of D belongs to at most a finite number of maximal t-ideals of D. AGCD domains of finite t-character were characterized in [34]. An ideal A of D is t-locally principal if AD _P is principal for every maximal t-ideal P of D. In [111] it was shown that if D is a PvMD, then D is of finite t-character if and only if every nonzero t-locally principal ideal of D is t-invertible. A GCD domain is a PvMD because the v-closure of every nonzero finitely generated ideal in a GCD domain is principal. We can therefore conclude that a GCD domain is of finite t-character if and only if every nonzero t-locally principal ideal A of D is t-invertible.

Now, we are in a position to state an open problem:

Let D be an almost GCD domain such that every nonzero t-locally principal ideal is t-invertible. Is D of finite t-character?

Problem 36.

Let D be an integral domain, let K be the field of quotients of D, and let $\overline{\boldsymbol{F}}(D)$ denote the set of all the nonzero D-submodules of K.

Let $\boldsymbol{\mathcal{T}}$ denote a nonempty collection of overrings of D and, for any $T \in \boldsymbol{\mathcal{T}}$, let ⋆ _T be a semistar operation on T. An interesting question posed in [23, Problem 44] is the following:

Problem 36a.

Find conditions on $\boldsymbol{\mathcal{T}}$ and on the semistar operations ⋆ _T under which the semistar operation $\star _{\boldsymbol{\mathcal{T}}}$ on D defined by $E^{\boldsymbol{\star }_{\boldsymbol{\mathcal{T}}}}:=\bigcap \{(ET)^{\star _{T}}\mid T \in \boldsymbol{\mathcal{T}}\}$, for all $E \in \overline{\boldsymbol{F}}(D),$ is of finite type.

Note that, if $D =\bigcap \{ T\mid T \in \boldsymbol{\mathcal{T}}\}$ is locally finite and each ⋆ _T is a star operation on T of finite type, then Anderson in [3, Theorem 2] proved that $\boldsymbol{\star }_{\boldsymbol{\mathcal{T}}}$ is a star operation on D of finite type. Through the years, several partial answers to this question were given and they are mainly topological in nature. For example, in [42, Corollary 4.6], a description of when the semistar operation $\boldsymbol{\star }_{\boldsymbol{\mathcal{T}}}$ is of finite type was given when $\boldsymbol{\mathcal{T}}$ is a family of localizations of D and ⋆ _T is the identity semistar operation on T, for each $T \in \boldsymbol{\mathcal{T}}$. More recently, in [41], it was proved that if $\boldsymbol{\mathcal{T}}$ is a quasi-compact subspace of the (Zariski–Riemann) space of all valuation overrings of D (endowed with the Zariski topology) and ⋆ _T is the identity (semi)star operation on T, for each $T \in \boldsymbol{\mathcal{T}}$, then $\boldsymbol{\star }_{\boldsymbol{\mathcal{T}}}$ is of finite type. Another more natural way to see the problem stated above is the following.

Problem 36b.

Let $\boldsymbol{\mathcal{S}}$ be any nonempty collection of semistar operations on D and let $\wedge _{\boldsymbol{\mathcal{S}}}$ be the semistar operation defined by $E^{\boldsymbol{\wedge }_{\boldsymbol{\mathcal{S}}}}:=\bigcap \{ E^{\star }\mid \star \in \boldsymbol{\mathcal{S}}\}$ for all $E \in \overline{\boldsymbol{F}}(D).$ Find conditions on the set $\boldsymbol{\mathcal{S}}$ for the semistar operation $\wedge _{\boldsymbol{\mathcal{S}}}$ on D to be of finite type.

Note that it is not so difficult to show that the constructions of the semistar operations of the type $\boldsymbol{\star }_{\boldsymbol{\mathcal{T}}}$ and $\boldsymbol{\wedge }_{\boldsymbol{\mathcal{S}}}$ are essentially equivalent, in the sense that every semistar operation $\boldsymbol{\star }_{\boldsymbol{\mathcal{T}}}$ can be interpreted as one of the type $\boldsymbol{\wedge }_{\boldsymbol{\mathcal{S}}}$, and conversely.

Problem 37.

A finite-dimensional integral domain D is said to be a Jaffard domain if $\dim (D[X_{1},X_{2},\ldots,X_{n}]) = n +\dim (D)$ for all n ≥ 1 and, equivalently, if $\dim (D) =\dim _{v}(D)$, where $\dim (D)$ denotes the (Krull) dimension of D and $\dim _{v}(D)$ its valuative dimension (i.e., the supremum of dimensions of the valuation overrings of D). As this notion does not carry over to localizations, D is said to be a locally Jaffard domain if D _P is Jaffard for each prime ideal P of D. The class of (locally) Jaffard domains contains most of the well-known classes of (locally) finite-dimensional rings involved in dimension theory such as Noetherian domains, Prüfer domains, universally catenarian domains, and universally strong S(eidenberg) domains. It is an open problem to compute the dimension of polynomial rings over Krull domains in general. In this vein, Bouvier conjectured that “finite-dimensional Krull (or, more particularly, factorial) domains need not be Jaffard” [18, 43]. Bouvier’s conjecture makes sense beyond the Noetherian context. Explicit finite-dimensional non-Noetherian Krull domains are scarce in the literature and one needs to test them and their localizations as well for the Jaffard property. In [16], the authors scanned all known families of examples of non-Noetherian finite-dimensional Krull (or factorial) domains existing in the literature. They showed that all these examples—except two—are in fact locally Jaffard domains. The two exceptions are addressed below in the open Problems 37b and 37c. Bouvier’s conjecture is still elusively open and one may reformulate it in the following simple terms:

Problem 37a.

Is there a Krull (or, more particularly, factorial) domain D such that $1 +\dim (D) \gneqq \dim (D[X])$?

In [16], Bouchiba and Kabbaj examined David’s second construction described in [29] of a three-dimensional factorial domain which arises as an ascending union of three-dimensional polynomial rings J _n in three variables over a field k; namely, J = ⋃ J _n with $J_{n} = k[X,\beta _{n-1},\beta _{n}]$ for each positive integer n, where the variables β _n satisfy the following condition: For n ≥ 2, $\beta _{n} = \frac{-\beta _{n-1}^{s(n)}+\beta _{ n-2}} {X}$, where the s(n) are positive integers. We have $J_{n} \subseteq J \subseteq J_{n}[X^{-1}]$ for each positive integer n. Therefore, by [31, Theorem 2.3], J is a Jaffard domain (since the J _n ’s are affine domains, i.e., finitely generated k-algebras), but it is not known if J is locally Jaffard. So the following question is open:

Problem 37b.

Is J a locally Jaffard domain?

Clearly, a negative answer to Problem 37b will solve (affirmatively) Bouvier’s conjecture for factorial domains (cf. Problem 37a).

In [16], the authors also investigated the known family of examples which stem from the generalized 14th problem of Hilbert (also called Hilbert–Zariski problem): Let k be a field of characteristic zero, T a normal affine domain (i.e., an integrally closed domain which is a finitely generated algebra) over k, and F a subfield of the quotient field of T. The Hilbert–Zariski problem asks whether D = F ∩ T is an affine domain over k. Counterexamples for this problem were constructed by Rees [100], Nagata [94], and Roberts [101], where D wasn’t even Noetherian. In this vein, Anderson et al. [8] asked whether D and its localizations inherit from T the Noetherian-like main behavior of having Krull and valuative dimensions coincide (i.e., whether D is locally Jaffard). In [18], the authors proved that D is Jaffard, but were not able to determine whether D is locally Jaffard. In fact, they addressed this problem in the more general context of subalgebras of affine domains over Noetherian domains and the following question remains open:

Problem 37c.

Let $A \subseteq D$ be an extension of integral domains, where A is Noetherian domain and D is a subalgebra of an affine domain T over A. Is D a locally Jaffard domain?

Clearly, a negative answer to Problem 37c will solve (affirmatively) Bouvier’s conjecture for Krull domains.

Problem 38.

A one-dimensional local Mori domain R is called locally tame if for every irreducible element u ∈ R there is a constant $t \in \mathbb{N}_{0}$ with the following property: For every a ∈ uR and every factorization $v_{1} \cdot \ldots \cdot v_{m}$ of a (where $m \in \mathbb{N}$ and $v_{1},\ldots,v_{m}$ are irreducibles of R), there is a subproduct, say $v_{1} \cdot \ldots \cdot v_{l}$, which is a multiple of u and which has a factorization containing u, say $v_{1} \cdot \ldots \cdot v_{l} = uu_{2} \cdot \ldots \cdot u_{k}$ (where $k \in \mathbb{N}_{0}$ and $u_{2},\ldots,u_{k}$ are irreducibles of R), such that max{k, l} ≤ t.

Let R be a one-dimensional local Mori domain. Is R locally tame?

Denote by $\hat{R}$ the complete integral closure of R. It is known that R is locally tame in each of the following cases: The conductor $(R\!:\!\hat{ R})\neq \{0\}$ [56, Proposition 2.10.7]; R is Noetherian [73, Theorem 3.3]; $\hat{R}$ is Krull and $\vert \mathfrak{X}(\hat{R})\vert \geq 2$ [58, Theorem 3.5 and Corollary 3.6].

Assume that R is locally tame. Then this implies the finiteness of several arithmetical invariants, such as the catenary degree and the set of distances of R. Also, the sets of lengths in R have a well-defined structure [55], [57], and [56, Theorem 3.1.1].

Problem 39.

Analyze and describe non-unique factorization in Int(D), where D is a DVR with finite residue field.

We remark that the results obtained by Frisch [47] for Int$(\mathbb{Z})$ rely heavily on the fact that $\mathbb{Z}$ has prime ideals of arbitrarily large index.

Problem 40.

Let ℤ (respectively, ℚ) denote the integers (respectively, the rationals), and let F be a field. It is known that Spec(F[x, y]) is order-isomorphic to $\mathrm{Spec}(\mathbb{Z}[y])$ if and only if F is contained in the algebraic closure of a finite field [109, Theorem 2.10]; and in that case $\mathrm{Spec}(A)$ is order-isomorphic to $\mathrm{Spec}(\mathbb{Z}[y])$ for every two-dimensional domain that is finitely generated as an F-algebra. Moreover, the poset $\mathrm{Spec}(\mathbb{Z}[y])$ is characterized, among posets, by five specific axioms [21, Theorem 2.9]. For more background information, see references [21, 108–110].

Problem 40a.

Find axioms characterizing the poset $\mathbb{Q}[x,y]$.

See [21, Remark 2.11.3].

Problem 40b.

Are $\mathrm{Spec}(\mathbb{Q}[x,y])$ and $\mathrm{Spec}(\mathbb{Q}(\sqrt{2})[x,y])$ order-isomorphic?

Problem 40c.

More generally, let F and K be algebraic number fields. If A is a two-dimensional affine domain over F, is Spec(A) order-isomorphic to Spec(K[x, y])?

This question is close to Question 2.15.1 of [110]. It is also observed in Example 2.14 of [110] and Corollary 7 of [108] that if L is an algebraically closed field of infinite transcendence degree over $\mathbb{Q}$, then the spectra of $L[x,y,z]/(x^{4} + y^{4} + z^{4} - 1)$ and L[x, y] are known to be non-isomorphic.

Problem 40d.

At the other extreme, let F and K be fields, neither of them algebraic over a finite field. If Spec(F[x, y] and Spec(K[x, y]) are order-isomorphic, are F and K necessarily isomorphic fields?

This is Question 2 of [110].

Problem 41.

Let I be the integral closure of the ideal $(x^{a},y^{b},z^{c})$ in a polynomial ring in variables x, y, z over a field.

Classify all triples (a, b, c) for which all powers of I are integrally closed.

For basic information on the integral closure of monomial ideals, see [81, Chap. 1, Sect. 1.4]. The case of two variables is known by Zariski’s theory of integral closure of ideals in a two-dimensional regular ring [81, Chap. 14], and for three variables some work was done by Coughlin in [27].

Problem 42.

For some special ideals, such as for monomial ideals, there are fast algorithms for computing their integral closure. For computing the integral closure of general ideals, however, the current algorithms reduce to computing the integral closure of the Rees algebra and then reading off the graded components of the integral closure This computes simultaneously the integral closures of all powers of the ideal, which is doing more than necessary. This excess of work makes the computation sometimes unwieldy.

Problem 42a.

Is there a more direct algorithm for computing the integral closure of a general ideal?

Problem 42b.

In particular, can the Leonard–Pelikaan [86] and Singh–Swanson [104] algorithms be modified for computing the integral closure of ideals?

More information and references on the topic may be found in [105, Sect. 2].

Problem 43.

The following is a general form of the Lipman–Sathaye Theorem, as found in Theorem 2.1 of [75]: Let R be a Noetherian domain with field of fractions K. Assume that the S ₂-locus is open in all algebras essentially of finite type over R. Let S be an extension algebra essentially of finite type over R such that S is torsion-free and generically étale over R, and such that for every maximal ideal M of S, R _M∩R is normal, and S _M has a relatively S ₂-presentation over R _M∩R . Let L = K ⊗ _R S and let $\overline{S}$ be the integral closure of S in L. Assume that $\overline{S}$ is module-finite over S, and that for every height-one prime ideal Q of $\overline{S}$, R _Q∩R is regular. Then $(\overline{S}: _{L}J_{\overline{S}/R}) \subseteq (S: _{L}J_{S/R}).$

The suggested problem is to find how tight is the statement of this theorem: relax some assumption and either prove the theorem or find a counterexample for that relaxation.

Problem 44.

Let R be a ring and let I be an ideal in R. A set of Rees valuation rings of I is a set $\{V _{1},..,V _{r}\}$ consisting of valuation rings, subject to the following conditions:

• (1) Each V _i is Noetherian and is not a field.

• (2) For each i = 1, …, r there exists a minimal prime ideal P _i of R such that V _i is a ring between R∕P _i and the field of fractions of R∕P _i .

• (3) For each natural number n, $\overline{I^{n}}$ = $\cap _{i=1}^{r}(I^{n}V _{i}) \cap R,$ where $\overline{I^{n}}$ denotes the integral closure of I ⁿ.

• (4) No set of valuation rings of cardinality smaller than r satisfies conditions(1)–(3).

Basic information on Rees valuations of ideals may be found in [81, Chap. 10]. Ideals that have only one Rees valuation have several good properties.

Cutkosky proved in [28] the existence of a Noetherian, two-dimensional, complete, integrally closed local domain (R, m) in which every m-primary ideal has more than one Rees valuation. Give a construction of such a ring.