arXiv subjects

Stefania Gabelli

Publications and source records attributed to Stefania Gabelli.

13 recordsLinked to original sources

arXiv2020

On finitely stable domains

We study Archimedean and locally Archimedean stable domains. We prove that a domain is stable and one-dimensional if and only if it is finitely stable and Mori. But we give examples of Archimedean stable local domains that are not one-dimensional. We also prove that a locally Archimedean stable domain satisfies accp and that Archimedean stable semilocal domains are locally Archimedean. But generally, neither Archimedean stable domains, nor Archimedean semilocal domains are necessarily locally Archimedean.

math.AC↗

arXiv2013

w-Stability and Clifford w-Regularity of Polynomial Rings

We investigate the transfer of w-stability and Clifford w-regularity from a domain D to the polynomial ring D[X]. We show that these two properties pass from D to D[X] when D is either integrally closed or it is Mori and w-divisorial.

math.AC↗

arXiv2013

Locally principal ideals and finite character

It is well-known that if R is a domain with finite character, each locally principal nonzero ideal of R is invertible. We address the problem of understanding when the converse is true and survey some recent results.

math.AC↗

arXiv2011

Stability and Clifford regularity with respect to star operations

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.

math.AC↗

arXiv2011

Star Stability and Star Regularity for Mori Domains

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.

math.AC↗

arXiv2008

Unique representation domains, II

Given a star operation * of finite type, we call a domain R a *-unique representation domain (*-URD) if each *-invertible *-ideal of R can be uniquely expressed as a *-product of pairwise *-comaximal ideals with prime radical. When * is the t-operation we call the *-URD simply a URD. Any unique factorization domain is a URD. Generalizing and unifying results due to Zafrullah and Brewer-Heinzer, we give conditions for a *-ideal to be a unique *-product of pairwise *-comaximal ideals with prime radical and characterize *-URDs. We show that the class of URDs includes rings of Krull type, the generalized Krull domains introduced by El Baghdadi and weakly Matlis domains whose t-spectrum is treed. We also study when the property of being a URD extends to some classes of overrings, such as polynomial extensions, rings of fractions and rings obtained by the D+XD_S[X] construction.

math.AC↗

arXiv2008

w-Divisoriality in Polynomial Rings

We extend the Bass-Matlis characterization of local Noetherian divisorial domains to the non-Noetherian case. This result is then used to study the following question: If a domain D is w-divisorial, that is, if each w-ideal of D is divisorial, then is D[X] automatically w-divisorial? We show that the answer is yes if D is either integrally closed or Mori.

math.AC↗

arXiv2006

Ring-theoretic properties of PVMDs

We extend to Prüfer $v$-multiplication domains some distinguished ring-theoretic properties of Prüfer domains. In particular we consider the $t##$-property, the $t$-radical trace property, $w$-divisoriality and $w$-stability.

math.AC↗

arXiv2006

Star Stable Domains

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.

math.AC↗

arXiv2004

w-Divisorial Domains

We study the class of domains in which each w-ideal is divisorial, extending several properties of divisorial and totally divisorial domains to a much wider class of domains. In particular we consider PvMDs and Mori domains.

math.AC↗

arXiv2004

The t#-property for intergral domains

We study a condition on intersections of localizations of a domain at maximal t-ideals. This extends and generalizes earlier work of Gilmer (1967), Gilmer-Heinzer (1968), Olberding (1998), and others for Prufer domains.

math.AC↗

arXiv2003

Maximal divisorial ideals and t-maximal ideals

We give conditions for a maximal divisorial ideal to be t-maximal and show with examples that, even in a completely integrally closed domain, maximal divisorial ideals need not be t-maximal.

math.AC↗

arXiv2003

Complete integral closure and strongly divisorial prime ideals

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

math.AC↗