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Marco Fontana

Publications and source records attributed to Marco Fontana.

At least 37 records · Page 2Linked to original sources
arXiv2010

An overring-theoretic approach to polynomial extensions of star and semistar operations

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 extensions) of semistar operations on $D$.

math.AC↗
arXiv2010

Properties of chains of prime ideals in an amalgamated algebra along an ideal

Let $f:A \to B$ be a ring homomorphism and let $J$ be an ideal of $B$. In this paper, we study the amalgamation of $A$ with $B$ along $J$ with respect to $f$ (denoted by ${A\Join^fJ}$), a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced and studied by D'Anna and Fontana in 2007, and other classical constructions (such as the $A+ XB[X]$, the $A+ XB[[X]]$ and the $D+M$ constructions). In particular, we completely describe the prime spectrum of the amalgamated duplication and we give bounds for its Krull dimension.

math.AC↗
arXiv2009

On $v$-domains: a survey

An integral domain $D$ is a $v$--domain if, for every finitely generated nonzero (fractional) ideal $F$ of $D$, we have $(FF^{-1})^{-1}=D$. The $v$--domains generalize Prüfer and Krull domains and have appeared in the literature with different names. This paper is the result of an effort to put together information on this useful class of integral domains. In this survey, we present old, recent and new characterizations of $v$--domains along with some historical remarks. We also discuss the relationship of $v$--domains with their various specializations and generalizations, giving suitable examples.

math.AC↗
arXiv2009

A "$v$-operation free" approach to Prüfer $v$-multiplication domains

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.

math.AC↗
arXiv2009

Cancellation properties in ideal systems: A classification of $\boldsymbol{e.a.b.}$ semistar operations

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 particular, we solve the open problem of showing that {\texttt{a.b.}} is really a stronger condition than {\texttt{e.a.b.}}

math.AC↗
arXiv2009

Amalgamated algebras along an ideal

Let $f:A \to B$ be a ring homomorphism and $J$ an ideal of $B$. In this paper, we initiate a systematic study of a new ring construction called the "amalgamation of $A$ with $B$ along $J$ with respect to $f$". This construction finds its roots in a paper by J.L. Dorroh appeared in 1932 and provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced and studied by D'Anna and Fontana in 2007, and other classical constructions such as the $A+ XB[X]$ and $A+ XB[[X]]$ constructions, the CPI-extensions of Boisen and Sheldon, the $D+M$ constructions and the Nagata's idealization.

math.AC↗
arXiv2008

Toward a classification of prime ideals in Prüfer domains

The primary purpose of this paper is give a classification scheme for the nonzero primes of a Prüfer domain based on five properties. A prime $P$ of a Prüfer domain $R$ could be sharp or not sharp, antesharp or not, divisorial or not, branched or unbranched, idempotent or not. Based on these five basic properties, there are six types of maximal ideals and twelve types of nonmaximal (nonzero) primes. Both characterizations and examples are given for each type that exists.

math.AC↗
arXiv2008

On $v$--domains and star operations

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, Academic Press, New York--London, 1971], which gives several equivalent conditions for an integral domain to be a Prüfer domain, as a model, and we show which statements of that theorem on Prüfer domains can be generalized in a natural way and proved for $\ast$--Prüfer domains, and which cannot be. We also show that in a $\ast $--Prüfer domain, each pair of $\ast $-invertible $\ast $-ideals admits a GCD in the set of $\ast $-invertible $\ast $-ideals, obtaining a remarkable generalization of a property holding for the "classical" class of Prüfer $v$--multiplication domains. We also link $D$ being $\ast $--Prüfer (or $(\ast, v)$--Prüfer) with the group Inv$^{\ast}(D)$ of $\ast $-invertible $\ast $-ideals (under $\ast$-multiplication) being lattice-ordered.

math.AC↗
arXiv2008

Uppers to zero in polynomial rings and Prüfer-like domains

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 semistar operation setting and studied in a unified frame. In this paper, given a semistar operation $\star$ in the sense of Okabe-Matsuda, we introduce the $\star$-quasi-Prüfer domains. We give several characterizations of these domains and we investigate their relations with the UM$t$-domains and the Prüfer $v$-multiplication domains.

math.AC↗
arXiv2007

Some remarks on Prüfer $\star $--multiplication domains and class groups

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 recognizing P$v$MD's which are not P$\ast $MD's for a certain finite type star operation $\ast $. We study domains $D$ for which the $\ast $--class group $\Cl^{\ast}(D)$ equals the $t$--class group $\Cl^{t}(D)$ for any finite type star operation $\ast $, and we indicate examples of P$v$MD's $D$ such that $\Cl^{\ast}(D)\subsetneq \Cl^{t}(D)$. We also compute $\Cl^v(D)$ for certain valuation domains $D$.

math.AC↗
arXiv2007

Uppers to zero and semistar operations in polynomial rings

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 and only if $D[X]$ is a Prüfer $[\star]$-multiplication (resp., a $[\star]$-Noetherian; a $[\star]$-Dedekind) domain. As an application of the techniques introduced here, we obtain a new interpretation of the Gabriel-Popescu localizing systems of finite type on an integral domain $D$ (Problem 45 of \cite{cg}), in terms of multiplicatively closed sets of the polynomial ring $D[X]$.

math.AC↗
arXiv2007

An amalgamated duplication of a ring along an ideal: the basic properties

We introduce a new general construction, denoted by $R\JoinE$, called the amalgamated duplication of a ring $R$ along an $R$--module $E$, that we assume to be an ideal in some overring of $R$. (Note that, when $E^2 =0$, $R\JoinE$ coincides with the Nagata's idealization $R\ltimes E$.) After discussing the main properties of the amalgamated duplication $R\JoinE$ in relation with pullback--type constructions, we restrict our investigation to the study of $R\JoinE$ when $E$ is an ideal of $R$. Special attention is devoted to the ideal-theoretic properties of $R\JoinE$ and to the topological structure of its prime spectrum.

math.AC↗
arXiv2006

The amalgamated duplication of a ring along a multiplicative-canonical ideal

After recalling briefly the main properties of the amalgamated duplication of a ring $R$ along an ideal $I$, denoted by $R\JoinI$, we restrict our attention to the study of the properties of $R\JoinI$, when $I$ is a multiplicative canonical ideal of $R$ \cite{hhp}. In particular, we study when every regular fractional ideal of $R\Join I$ is divisorial.

math.AC↗
arXiv2006

Factoring Ideals in Prüfer Domains

We show that in certain Prüfer domains, each nonzero ideal $I$ can be factored as $I=I^v Π$, where $I^v$ is the divisorial closure of $I$ and $Π$ is a product of maximal ideals. This is always possible when the Prüfer domain is $h$-local, and in this case such factorizations have certain uniqueness properties. This leads to new characterizations of the $h$-local property in Prüfer domains. We also explore consequences of these factorizations and give illustrative examples.

math.AC↗
arXiv2006

Prüfer $\star$--multiplication domains and $\star$--coherence

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 semistar setting the notion of $\texttt{H}$--domain introduced by Glaz and Vasconcelos and we show, among the other results that, in the class of the $\texttt{H}(\star)$--domains, the Prüfer $\star$--multiplication domains coincide with the $\star$--domains.

math.AC↗
arXiv2006

A generalization of Kronecker function rings and Nagata rings

Despite the obvious similarities in definitions and properties, the Nagata ring and the Kronecker function ring have been classically treated independently, when the coefficients are not in a Prüfer domain. The purpose of this paper is to study two different unified approaches to the Nagata rings and the Kronecker function rings, which yield these rings and their classical generalizations as special cases.

math.AC↗