In this paper we study the star operations on a pullback of integral domains. In particular, we characterize the star operations of a domain arising from a pullback of ``a general type'' by introducing new techniques for ``projecting'' and ``lifting'' star operations under surjective homomorphisms of integral domains. We study the transfer in a pullback (or with respect to a surjective homomorphism) of some relevant classes or distinguished properties of star operations such as $v-, t-, w-, b-, d-,$ finite type, e.a.b., stable, and spectral operations. We apply part of the theory developed here to give a complete positive answer to a problem posed by D. F. Anderson in 1992 concerning the star operations on the ``$D+M$'' constructions.
For the domain $R$ arising from the construction $T, M,D$, we relate the star class groups of $R$ to those of $T$ and $D$. More precisely, let $T$ be an integral domain, $M$ a nonzero maximal ideal of $T$, $D$ a proper subring of $k:=T/M$, $ϕ: T\to k$ the natural projection, and let $R=ϕ^{-1}(D)$. For each star operation $\ast$ on $R$, we define the star operation $\ast_ϕ$ on $D$, i.e., the ``projection'' of $\ast$ under $ϕ$, and the star operation ${(\ast)}_{_{T}}$ on $T$, i.e., the ``extension'' of $\ast$ to $T$. Then we show that, under a mild hypothesis on the group of units of $T$, if $\a…
For a finite-type star operation $\star$ on a domain $R$, we say that $R$ is $\star$-super potent if each maximal $\star$-ideal of $R$ contains a finitely generated ideal $I$ such that (1) $I$ is contained in no other maximal $\star$-ideal of $R$ and (2) $J$ is $\star$-invertible for every finitely generated ideal $J \supseteq I$. Examples of $t$-super potent domains include domains each of whose maximal $t$-ideals is $t$-invertible (e.g., Krull domains). We show that if the domain $R$ is $\star$-super potent for some finite-type star operation $\star$, then $R$ is $t$-super potent, we study $…
Let $R$ be an integral domain, $Star(R)$ the set of all star operations on $R$ and $StarFC(R)$ the set of all star operations of finite type on $R$. Then $R$ is said to be star regular if $|Star(T)|\leq |Star(R)|$ for every overring $T$ of $R$. In this paper we introduce the notion of finitely star regular domain as an integral domain $R$ such that $|StarFC(T)|\leq |StarFC(R)|$ for each overring $T$ of $R$. First, we show that the notions of star regular and finitely star regular domains are completely different and do not imply each other. Next, we extend/generalize well-known results on star…
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 show that Krull's \texttt{a.b.} cancellation condition is a properly stronger condition than Gilmer's \texttt{e.a.b.} cancellation condition for star operations.
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…
We give a classification of {\texttt{e.a.b.}} semistar (and star) operations by defining four different (successively smaller) distinguished classes. Then, using a standard notion of equivalence of semistar (and star) operations to partition the collection of all {\texttt{e.a.b.}} semistar (or star) operations, we show that there is exactly one operation of finite type in each equivalence class and that this operation has a range of nice properties. We give examples to demonstrate that the four classes of {\texttt{e.a.b.}} semistar (or star) operations we defined can all be distinct. In partic…
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 $\star_f$-…
The so called Prüfer $v$-multiplication domains (P$v$MD's) are usually defined as domains whose finitely generated nonzero ideals are $t$-invertible. These domains generalize Prüfer domains and Krull domains. The P$v$MD's are relatively obscure compared to their very well known special cases. One of the reasons could be that the study of P$v$MD's uses the jargon of star operations, such as the $v$-operation and the $t$-operation. In this paper, we provide characterizations of and basic results on P$v$MD's and related notions without star operations.
We generalize the concept of localization of a star operation to flat overrings; subsequently, we investigate the possibility of representing the set $\mathrm{Star}(R)$ of star operations on $R$ as the product of $\mathrm{Star}(T)$, as $T$ ranges in a family of overrings of $R$ with special properties. We then apply this method to study the set of star operations on a Prüfer domain $R$, in particular the set of stable star operations and the star-class groups of $R$.
Let $\ast$ be a star operation on an integral domain $D$. Let $\f(D)$ be the set of all nonzero finitely generated fractional ideals of $D$. Call $D$ a $\ast$--Prüfer (respectively, $(\ast, v)$--Prüfer) domain if $(FF^{-1})^{\ast}=D$ (respectively, $(F^vF^{-1})^{\ast}=D$) for all $F\in \f(D)$. We establish that $\ast$--Prüfer domains (and $(\ast, v)$--Prüfer domains) for various star operations $\ast $ span a major portion of the known generalizations of Prüfer domains inside the class of $v$--domains. We also use Theorem 6.6 of the Larsen and McCarthy book [Multiplicative Theory of Ideals, Ac…
We study star operations on Kunz domains, a class of analytically irreducible, residually rational domains associated to pseudo-symmetric numerical semigroups, and we use them to refute a conjecture of Houston, Mimouni and Park. We also find an estimate for the number of star operations in a particular case, and a precise counting in a sub-case.
In the last few years, the concepts of stability and Clifford regularity have been fruitfully extended by using star operations. In this paper we deepen the study of star stable and star regular domains and relate these two classes of domains to each other.
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…
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.
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.
We study the set of star operations on local Noetherian domains $D$ of dimension $1$ such that the conductor $(D:T)$ (where $T$ is the integral closure of $D$) is equal to the maximal ideal of $D$. We reduce this problem to the study of a class of closure operations (more precisely, multiplicative operations) in a finite extension $k\subseteq B$, where $k$ is a field, and then we study how the cardinality of this set of closures vary as the size of $k$ varies while the structure of $B$ remains fixed.
Let $D$ be an integral domain with quotient field $K$. A star-operation $\star$ on $D$ is a closure operation $A \longmapsto A^\star$ on the set of nonzero fractional ideals, $F(D)$, of $D$ satisfying the properties: $(xD)^\star = xD$ and $(xA)^\star = xA^\star$ for all $x \in K^\ast$ and $A \in F(D)$. Let ${\M S}$ be a multiplicatively closed set of ideals of $D$. For $A \in F(D)$ define $A_{\M S} = \{x \in K \mid xI \subseteq{A}$, for some $I \in {\M S}\}$. Then $D_{\M S}$ is an overring of $D$ and $A_{\M S}$ is a fractional ideal of $D_{\M S}$. Let ${\M S}$ be a multiplicative set of finite…
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 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…
We introduce an order on the set of non-divisorial ideals of a numerical semigroup $S$, and link antichains of this order with the star operations on $S$; subsequently, we use this order to find estimates on the number of star operations on $S$. We then use them to find an asymptotic estimate on the number of nonsymmetric numerical semigroups with $n$ or less star operations, and to determine these semigroups explicitly when $n=10$.
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 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 …
This paper seeks ring-theoretic conditions of an integral domain R that reflect in the Clifford property or Boolean property of its class semigroup S(R), that is, the semigroup of the isomorphy classes of the nonzero (integral) ideals of R with the operation induced by multiplication. Precisely, in Section 3, we characterize integrally closed domains with Boolean class semigoup; in this case, S(R) identifies with the Boolean semigroup formed of all fractional overrings of R. In Section 4, we investigate Noetherian-like settings where the Clifford and Boolean properties of S(R) coincide with (L…
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 …
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.
Let $A\subseteq B$ be a ring extension and $\mathcal{G}$ be a set of $A$-submodules of $B$. We introduce a class of closure operations on $\mathcal{G}$ (which we call \emph{multiplicative operations on $(A,B,\mathcal{G})$}) that generalizes the classes of star, semistar and semiprime operations. We study how the set $\mathrm{Mult}(A,B,\mathcal{G})$ of these closure operations vary when $A$, $B$ or $\mathcal{G}$ vary, and how $\mathrm{Mult}(A,B,\mathcal{G})$ behave under ring homomorphisms. As an application, we show how to reduce the study of star operations on analytically unramified one-dime…
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 $D_{P}$…
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…
Let R be a commutative ring and I an ideal of R. A sub-ideal J of I is a reduction of I if JI^n = I^n+1 for some positive integer n. The ring R has the (finite) basic ideal property if (finitely generated) ideals of R do not have proper reductions. Hays characterized (one-dimensional) Prufer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays' results to Prufer v-multiplication domains by replacing "basic" with "w-basic," where w is a particular star operation. We also investigate relations among star-basic properties for certain star operations.
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 …
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.
Let $D$ be an integral domain and $\star $ a star operation defined on $D$. We say that $D$ is a $\star $-power conductor domain ($\star $-PCD) if for each pair $a,b\in D\backslash (0)$ and for each positive integer $n$ we have $Da^{n}\cap Db^{n}=((Da\cap Db)^{n})^{\ast }.$ We study $\star $-PCDs and characterize them as root closed domains satisfying $ ((a,b)^{n})^{-1}=(((a,b)^{-1})^{n})^{\star }$ for all nonzero $a,b$ and all natural numbers $n\geq 1$. From this it follows easily that Prüfer domains are $d$-PCDs (where $d$ denotes the trivial star operation), and $v$ -domains (e.g., Krull do…
In the last few years, the concepts of stability and Clifford regularity have been fruitfully extended by using star operations. In this paper we study and put in relation these properties for Noetherian and Mori domains, substantially improving several results present in the literature.
