Let $D$ be an integral domain and $\star$ a semistar operation on $D$. As a generalization of the notion of Noetherian domains to the semistar setting, we say that $D$ is a $\star$--Noetherian domain if it has the ascending chain condition on the set of its quasi--$\star$--ideals. On the other hand, as an extension the notion of Prüfer domain (and of Prüfer $v$--multiplication domain), we say that $D$ is a Prüfer $\star$--multiplication domain (P$\star$MD, for short) if $D_M$ is a valuation domain, for each quasi--$\star_{_{f}}$--maximal ideal $M$ of $D$. Finally, recalling that a Dedekind domain is a Noetherian Prüfer domain, we define a $\star$--Dedekind domain to be an integral domain which is $\star$--Noetherian and a P$\star$MD. In the present paper, after a preliminary study of $\star$--Noetherian domains, we investigate the $\star$--Dedekind domains. We extend to the $\star$--Dedekind domains the main classical results and several characterizations proven for Dedekind domains. In particular, we obtain a characterization of a $\star$--Dedekind domain by a property of decomposition of any semistar ideal into a ``semistar product'' of prime ideals. Moreover, we show that an integral domain $D$ is a $\star$--Dedekind domain if and only if the Nagata semistar domain Na$(D, \star)$ is a Dedekind domain. Several applications of the general results are given for special cases of the semistar operation $\star$.
We give an explicit description of the lattice $\Semistar(D)$ of all semistar operations on any Dedekind domain $D$ from its set $\Max(D)$ of maximal ideals. This descpription is constructive if $\Max(D)$ is finite. As a corollary we show that $2^{n \choose [n/2]} \leq |\Semistar(D)| \leq 2^{2^n}$ if $n = |\Max(D)|$ is finite; we compute $|\Semistar(D)|$ if $|\Max(D)| \leq 7$; and we show that if $\Max(D)$ is infinite then $\Semistar(D)$ has cardinality $2^{2^{|\Max(D)|}}$.
Given a stable semistar operation of finite type $\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $[\star]$ on the polynomial ring $D[X]$, such that $D$ is a $\star$-quasi-Prüfer domain if and only if each upper to zero in $D[X]$ is a quasi-$[\star]$-maximal ideal. This result completes the investigation initiated by Houston-Malik-Mott \cite[Section 2]{hmm} in the star operation setting. Moreover, we show that $D$ is a Prüfer $\star$-multiplication (resp., a $\star$-Noetherian; a $\star$-Dedekind) domain if a…
After the introduction in 1994, by Okabe and Matsuda, of the notion of semistar operation, many authors have investigated different aspects of this general and powerful concept. A natural development of the recent work in this area leads to investigate the concept of invertibility in the semistar setting. In this paper, we will show the existence of a ``theoretical obstruction'' for extending many results, proved for star-invertibility, to the semistar case. For this reason, we will introduce two distinct notions of invertibility in the semistar setting (called $\star$--invertibility and quasi…
In 1994, Matsuda and Okabe introduced the notion of semistar operation, extending the "classical" concept of star operation. In this paper, we introduce and study the notions of semistar linkedness and semistar flatness which are natural generalizations, to the semistar setting, of their corresponding "classical" concepts. As an application, among other results, we obtain a semistar version of Davis' and Richman's overring-theoretical theorems of characterization of Prüfer domains for Prüfer semistar multiplication domains.
Call a semistar operation $\ast$ on the polynomial domain $D[X]$ an extension (respectively, a strict extension) of a semistar operation $\star$ defined on an integral domain $D$, with quotient field $K$, if $E^\star = (E[X])^{\ast}\cap K$ (respectively, $E^\star [X]= (E[X])^{\ast}$) for all nonzero $D$-submodules $E$ of $K$. In this paper, we study the general properties of the above defined extensions and link our work with earlier efforts, centered on the stable semistar operation case, at defining semistar operations on $D[X]$ that are "canonical" extensions (or, "canonical" strict extensi…
We study stable semistar operations defined over a Prüfer domain, showing that, if every ideal of a Prüfer domain $R$ has only finitely many minimal primes, every such closure can be described through semistar operations defined on valuation overrings of $R$.
We study the sets of semistar and star operation on a semilocal Prüfer domain, with an emphasis on which properties of the domain are enough to determine them. In particular, we show that these sets depend chiefly on the properties of the spectrum and of some localizations of the domain; we also show that, if the domain is $h$-local, the number of semistar operations grows as a polynomial in the number of semistar operations of its localizations.
Let $D$ be an integral domain and $\star$ a semistar operation stable and of finite type on it. In this paper we define the semistar dimension (inequality) formula and discover their relations with $\star$-universally catenarian domains and $\star$-stably strong S-domains. As an application we give new characterizations of $\star$-quasi-Prüfer domains and UM$t$ domains in terms of dimension inequality formula (and the notions of universally catenarian domain, stably strong S-domain, strong S-domain, and Jaffard domains). We also extend Arnold's formula to the setting of semistar operations.
In this note we present a corregindium to our paper "Krull dimension, Overrings and Semistar operations of an integral Domain", Journal of Algebra, 321 (2009), 1497--1509.
We study the "local" behavior of several relevant properties concerning semistar operations, like finite type, stable, spectral, e.a.b. and a.b. We deal with the "global" problem of building a new semistar operation on a given integral domain, by "gluing" a given homogeneous family of semistar operations defined on a set of localizations. We apply these results for studying the local--global behavior of the semistar Nagata ring and the semistar Kronecker function ring. We prove that an integral domain $D$ is a Pr{ü}fer $\star$--multiplication domain if and only if all its localizations $D_{P}$…
Let $\star$ be a semistar operation on a domain $D$, $\star_f$ the finite-type semistar operation associated to $\star$, and $D$ a Prüfer $\star$-multiplication domain (P$\star$MD). For the special case of a Prüfer domain (where $\star$ is equal to the identity semistar operation), we show that a nonzero prime $P$ of $D$ is sharp, that is, that $D_P \nsupseteq \bigcap D_M$, where the intersection is taken over the maximal ideals $M$ of $D$ that do not contain $P$, if and only if two closely related spectral semistar operations on $D$ differ. We then give an appropriate definition of $\star_f$-…
Let $D$ be an integral domain and $\star$ a semistar operation stable and of finite type on it. In this paper, we are concerned with the study of the semistar (Krull) dimension theory of polynomial rings over $D$. We introduce and investigate the notions of $\star$-universally catenarian and $\star$-stably strong S-domains and prove that, every $\star$-locally finite dimensional Prüfer $\star$-multiplication domain is $\star$-universally catenarian, and this implies $\star$-stably strong S-domain. We also give new characterizations of $\star$-quasi-Prüfer domains introduced recently by Chang a…
We introduce and study the set of radical stable operations of an integral domain $D$. We show that their set is a complete lattice that is the join-completion of the set of spectral semistar operations, and we characterize when every radical operation is spectral (under the hypothesis that $D$ is rad-colon coherent). When $D$ is a Prüfer domain such that every set of minimal prime ideals is scattered, we completely classify stable semistar operations.
A class of integer-valued functions defined on the set of ideals of an integral domain $R$ is investigated. We show that this class of functions, which we call ideal valuations, are in one-to-one correspondence with countable descending chains of finite type, stable semistar operations with largest element equal to the $e$-operation. We use this class of functions to recover familiar semistar operations such as the $w$-operation and to give a solution to a conjecture by Chapman and Glaz when the ring is a valuation domain.
Given a stable semistar operation of finite type $\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $\star[X]$ on the polynomial ring $D[X]$, such that, if $n:=\star$-$\dim(D)$, then $n+1\leq \star[X]\text{-}\dim(D[X])\leq 2n+1$. We also establish that if $D$ is a $\star$-Noetherian domain or is a Prüfer $\star$-multiplication domain, then $\star[X]\text{-}\dim(D[X])=\star\text{-}\dim(D)+1$. Moreover we define the semistar valuative dimension of the domain $D$, denoted by $\star$-$\dim_v(D)$, to be the maximal …
We prove a characterization of a P$\star$MD, when $\star$ is a semistar operation, in terms of polynomials (by using the classical characterization of Prüfer domains, in terms of polynomials given by R. Gilmer and J. Hoffman \cite{Gilmer/Hoffmann:1974}, as a model), extending a result proved in the star case by E. Houston, S.J. Malik and J. Mott \cite{Houston/Malik/Mott:1984}. We also deal with the preservation of the P$\star$MD property by ``ascent'' and ``descent'' in case of field extensions. In this context, we generalize to the P$\star$MD case some classical results concerning Prüfer doma…
Let $R$ be a commutative integral domain and let $\star$ be a semistar operation of finite type on $R$, and $I$ be a quasi-$\star$-ideal of $R$. We show that, if every minimal prime ideal of $I$ is the radical of a $\star$-finite ideal, then the set $\Min(I)$ of minimal prime ideals over $I$ is finite.
Let $D$ be a Prüfer $\star$-multiplication domain, where $\star$ is a semistar operation on $D$. We show that certain ideal-theoretic properties related to idempotence and divisoriality hold in Prüfer domains, and we use the associated semistar Nagata ring of $D$ to show that the natural counterparts of these properties also hold in $D$.
Let $D$ be an integral domain with quotient field $K$. The $b$-operation that associates to each nonzero $D$-submodule $E$ of $K$, $E^b := \bigcap\{EV \mid V valuation overring of D\}$, is a semistar operation that plays an important role in many questions of ring theory (e.g., if $I$ is a nonzero ideal in $D$, $I^b$ coincides with its integral closure). In a first part of the paper, we study the integral domains that are $b$-Noetherian (i.e., such that, for each nonzero ideal $I$ of $D$, $I^b = J^b$ for some a finitely generated ideal $J$ of $D$). For instance, we prove that a $b$-Noetherian …
We show that a generalization of the theory of quantales and prequantales provides a noncommutative and nonassociative abstract ideal theoretic setting for the theories of star operations, semistar operations, semiprime operations, ideal systems, and module systems, and conversely the latter theories motivate new results in the theory of quantales and prequantales. Applications include representation theorems for precoherent prequantales and multiplicative semilattices; characterizations of simple commutative quantales and simple multiplicative lattices; a construction of the largest finitary …
Let $D$ be an integral domain and $X$ an indeterminate over $D$. It is well known that (a) $D$ is quasi-Prüfer (i.e, its integral closure is a Prüfer domain) if and only if each upper to zero $Q$ in $D[X] $ contains a polynomial $g \in D[X]$ with content $\co_D(g) = D$; (b) an upper to zero $Q$ in $D[X]$ is a maximal $t$-ideal if and only if $Q$ contains a nonzero polynomial $g \in D[X]$ with $\co_D(g)^v = D$. Using these facts, the notions of UM$t$-domain (i.e., an integral domain such that each upper to zero is a maximal $t$-ideal) and quasi-Prüfer domain can be naturally extended to the sem…
This is an extended introduction to discrete valuation rings and Dedekind domains. Some natural generalizations of Dedekind domains are also (briefly) discussed including "almost Dedekind domains", Prüfer domains, Krull domains, and singular integral domains of dimension 1.
We give an affirmative answer to a 1976 question of M. Rosen: every abelian group is isomorphic to the class group of an elliptic Dedekind domain R. We can choose R to be the integral closure of a PID in a separable quadratic field extension. In particular, this yields new and -- we feel -- simpler proofs of theorems of L. Claborn and C.R. Leedham-Green.
The purpose of this paper is to deepen the study of the Prüfer $\star$--multiplication domains, where $\star$ is a semistar operation. For this reason, in Section 2, we introduce the $\star$--domains, as a natural extension of the $v$--domains, where $v$ is the classical Artin's divisorial operation. We investigate their close relation with the Prüfer $\star$--multiplication domains. In particular, in Section 3, we obtain a characterization of Prüfer $\star$--multiplication domains in terms of $\star$--domains satisfying a variety of coherent-like conditions. In Section 4, we extend to the sem…
We study different form of boundness for ideals of almost Dedekind domains, generalizing the notions of critical ideals, radical factorization, and SP-domains. We show that every almost Dedekind domain has at least one noncritical maximal ideals and, indeed, the set of noncritical maximal ideals is dense in the maximal space, with respect to the constructible topology; as a consequence, we show that every almost Dedekind domain is SP-scattered, and in particular that the group $\mathrm{Inv}(D)$ of invertible ideals of an almost Dedekind domain $D$ is always free. If $D$ is an almost Dedekind d…
We introduce and study the notion of $\star$-stability with respect to a semistar operation $\star$ defined on a domain $R$; in particular we consider the case where $\star$ is the $w$-operation. This notion allows us to generalize and improve several properties of stable domains and totally divisorial domains.
Bhargava parametrized quintic rings over $\mathbb{Z}$ by quadruples of $5\times 5$ alternating matrices. We extend the construction to work similarly over any Dedekind domain $R$. No assumptions are needed on the characteristic of $R$. The resolvent consists of a pair of locally free modules $L$, $M$ with two multilinear maps between them; we can view $L$ as $Q/R$, for $Q$ the quintic ring, and $M$ as $S/R$, where $S$ is a sextic resolvent ring. As in Bhargava's treatment, any quintic ring has a resolvent ring, and for a maximal ring, the resolvent is unique. We hope that this work will enable…
For a Dedekind domain $D$, let $\mathcal{P}(D)$ be the set of ideals of $D$ that are radical of a principal ideal. We show that, if $D,D'$ are Dedekind domains and there is an order isomorphism between $\mathcal{P}(D)$ and $\mathcal{P}(D')$, then the rank of the class groups of $D$ and $D'$ is the same.
We study the Golomb spaces of Dedekind domains with torsion class group. In particular, we show that a homeomorphism between two such spaces sends prime ideals into prime ideals and preserves the $P$-adic topology on $R\setminus P$. Under certain hypothesis, we show that we can associate to a prime ideal $P$ of $R$ a partially ordered set, constructed from some subgroups of the group of units of $R/P^n$, which is invariant under homeomorphisms, and use this result to show that the unique self-homeomorphisms of the Golomb space of $\mathbb{Z}$ are the identity and the multiplication by $-1$. We…
We present a generalization of a polynomial factorization algorithm that works with ideals in maximal orders of global function fields. The method presented in this paper is intrinsic in the sense that it does not depend on the embedding of the ring of polynomials into the Dedekind domain in question.
Gröbner bases are a fundamental tool when studying ideals in multivariate polynomial rings. More recently there has been a growing interest in transferring techniques from the field case to other coefficient rings, most notably Euclidean domains and principal ideal rings. In this paper we will consider multivariate polynomial rings over Dedekind domain. By generalizing methods from the theory of finitely generated projective modules, we show that it is possible to describe Gröbner bases over Dedekind domains in a way similar to the case of principal ideal domains, both from a theoretical and a…
In 2001, M. Bhargava stunned the mathematical world by extending Gauss's 200-year-old group law on integral binary quadratic forms, now familiar as the ideal class group of a quadratic ring, to yield group laws on a vast assortment of analogous objects. His method yields parametrizations of rings of degree up to 5 over the integers, as well as aspects of their ideal structure, and can be employed to yield statistical information about such rings and the associated number fields. In this paper, we extend a selection of Bhargava's most striking parametrizations to cases where the base ring is no…
We study almost Dedekind domains with respect to the failure of ideals to have radical factorization, that is, we study how to measure how far an almost Dedekind domain is from being an SP-domain. To do so, we consider the maximal space $\mathcal{M}=\mathrm{Max}(R)$ of an almost Dedekind domain $R$, interpreting its (fractional) ideals as maps from $\mathcal{M}$ to $\mathbb{Z}$, and looking at the continuity of these maps when $\mathcal{M}$ is endowed with the inverse topology and $\mathbb{Z}$ with the discrete topology. We generalize the concept of critical ideals by introducing a well-ordere…
We prove that all pure submodules of a finite rank torsion-free module $A$ over a Dedekind domain are $A$-generated if and only if $A$ has a rank $1$ direct summand $B$ such that $\mathbf{type}(B)$ is the inner type of $A$.
For a Dedekind domain $\mathcal{O}$ and a rank two co-torsion module $M\subseteq \mathcal{O}^2$ with invariant factor ideals $\mathcal{L}\supseteq \mathcal{K}$ in $\mathcal{O}$, that is, $\frac{\mathcal{O}^2}{M}\cong \frac{\mathcal{O}}{\mathcal{L}}\oplus \frac{\mathcal{O}}{\mathcal{K}}$ we associate a new projective space invariant element in $\mathbb{PF}^1_{\mathcal{I}}$ where $\mathcal{I}$ is given by the ideal factorization $\mathcal{K} = \mathcal{L}\mathcal{I}$ in $\mathcal{O}$. This invariant element along with the invariant factor ideals determine the module $M$ completely as a subset of…
We introduce a new class of commutative rings with unity, namely, the Containment-Division Rings (CDR-s). We show that this notion has a very exceptional origin since it was essentially co-discovered with the qualitative help of a computer program (i.e. The Heterogeneous Tool Set (HETS)). Besides, we show that in a Noetherian setting, the CDR-s are just another way of describing Dedekind domains. Simultaneously, we see that for CDR-s, the Noetherian condition can be replaced by a weaker Divisor Chain Condition.
Let $D$ be a Dedekind domain with infinitely many maximal ideals, all of finite index, and $K$ its quotient field. Let $\operatorname{Int}(D) = \{f\in K[x] \mid f(D) \subseteq D\}$ be the ring of integer-valued polynomials on $D$. Given any finite multiset $\{k_1, \ldots, k_n\}$ of integers greater than $1$, we construct a polynomial in $\operatorname{Int}(D)$ which has exactly $n$ essentially different factorizations into irreducibles in $\operatorname{Int}(D)$, the lengths of these factorizations being $k_1$, \ldots, $k_n$. We also show that there is no transfer homomorphism from the multipl…
Let $R$ be an integral domain with $qf(R)=K$ and let $F(R)$ be the set of nonzero fractional ideals of $R.$ Call $R$ a dually compact domain (DCD) if for each $I\in F(R)$ the ideal $I_{v}=(I^{-1})^{-1}$ is a finite intersection of principal fractional ideals. We characterize DCDs and show that the class of DCDs properly contains various classes of integral domains, such as Noetherian, Mori and Krull domains. In addition we show that a Schreier DCD is a GCD domain with the property that for each $A\in F(R)$ the ideal $A_{v}$ is principal. We show that a domain $R$ is G-Dedekind domain (i.e. has…
Let $V$ be a minimal valuation overring of an integral domain $D$ and let $\mathrm{Zar}(D)$ be the Zariski space of the valuation overrings of $D$. Starting from a result in the theory of semistar operations, we prove a criterion under which the set $\mathrm{Zar}(D)\setminus\{V\}$ is not compact. We then use it to prove that, in many cases, $\mathrm{Zar}(D)$ is not a Noetherian space, and apply it to the study of the spaces of Kronecker function rings and of Noetherian overrings.
Kamiya, Takemura, and Terao initiated the theory of the characteristic quasi-polynomial of an integral arrangement, which is a function counting the elements in the complement of the arrangement modulo positive integers. They gave a period of the characteristic quasi-polynomial, called the LCM-period, and showed that the first constituent of the characteristic quasi-polynomial coincides with the characteristic polynomial of the corresponding hyperplane arrangement. Recently, Liu, Tran, and Yoshinaga showed that the last constituent of the characteristic quasi-polynomial coincides with the char…
Let $R=\bigoplus_{α\inΓ}R_α$ be a graded integral domain graded by an arbitrary grading torsionless monoid $Γ$, and $\star$ be a semistar operation on $R$. In this paper we define and study the graded integral domain analogue of $\star$-Nagata and Kronecker function rings of $R$ with respect to $\star$. We say that $R$ is a graded Prüfer $\star$-multiplication domain if each nonzero finitely generated homogeneous ideal of $R$ is $\star_f$-invertible. Using $\star$-Nagata and Kronecker function rings, we give several different equivalent conditions for $R$ to be a graded Prüfer $\star$-multipli…
Let $D$ be a Dedekind domain. Roughly speaking, a simultaneous $\mathfrak{p}$-ordering is a sequence of elements from $D$ which is equidistributed modulo every power of every prime ideal in $D$ as well as possible. Bhargava asked which subsets of the Dedekind domains admit simultaneous $\mathfrak{p}$-orderings. We give an overview on the progress in this problem. We also explain how it relates to the theory of integer valued polynomials and list some open problems.
Let $D$ be an integral domain with quotient field $K$ and let $X$ be an indeterminate over $D$. Also, let $\boldsymbol{\mathcal{T}}:=\{T_λ\mid λ \in Λ\}$ be a defining family of quotient rings of $D$ and suppose that $\ast $ is a finite type star operation on $D$ induced by $\boldsymbol{\mathcal{T}}$. We show that $D$ is a P$ \ast $MD (resp., P$v$MD) if and only if $(\co_D(fg))^{\ast}=(\co_D(f)\co_D(g))^{\ast}$ (resp., $(\co_D(fg))^{w}=(\co_D(f)\co_D(g))^{w}$) for all $0 \ne f,g \in K[X]$. A more general version of this result is given in the semistar operation setting. We give a method for re…
A pair of elements $a,b$ in an integral domain $R$ is an idempotent pair if either $a(1-a) \in bR$, or $b(1-b) \in aR$. $R$ is said to be a PRINC domain if all the ideals generated by an idempotent pair are principal. We show that in an order $R$ of a Dedekind domain every regular prime ideal can be generated by an idempotent pair; moreover, if $R$ is PRINC, then its integral closure, which is a Dedekind domain, is PRINC, too. Hence, a Dedekind domain is PRINC if and only if it is a PID. Furthermore, we show that the only imaginary quadratic orders $\mathbb Z[\sqrt{-d}]$, $d > 0$ square-free, …
We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gröbner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen-Macaulay domains that are Koszul, that …
Let $R=\bigoplus_{α\inΓ}R_α$ be a graded integral domain and $\star$ be a semistar operation on $R$. For $a\in R$, denote by $C(a)$ the ideal of $R$ generated by homogeneous components of $a$ and for$f=f_0+f_1X+\cdots+f_nX^n\in R[X]$, let $\A_f:=\sum_{i=0}^nC(f_i)$. Let $N(\star):=\{f\in R[X]\mid f\neq0\text{and}\A_f^{\star}=R^{\star}\}$. In this paper we study relationships between ideal theoretic properties of $\NA(R,\star):=R[X]_{N(\star)}$ and the homogeneous ideal theoretic properties of $R$. For example we show that $R$ is a graded Prüfer-$\star$-multiplication domain if and only if $\NA…
This paper studies the notions of star and semistar operations over a polynomial ring. It aims at characterizing when every upper to zero in $R[X]$ is a $*$-maximal ideal and when a $*$-maximal ideal $Q$ of $R[X]$ is extended from $R$, that is, $Q=(Q\cap R)[X]$ with $Q\cap R\not =0$, for a given star operation of finite character $*$ on $R[X]$. We also answer negatively some questions raised by Anderson-Clarke by constructing a Prüfer domain $R$ for which the $v$-operation is not stable.
