Warfield Domains

Bazzoni, Silvana; Salce, Luigi

doi:10.1006/jabr.1996.0353

Cited by 57 publications

(54 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The class of domains in which each nonzero ideal is divisorial was studied, independently and with different methods, by Bass [10], Matlis [27], and Heinzer [19] in the 1960s. Following Bazzoni and Salce [12,11], these domains are now called divisorial domains. Among other results, Heinzer proved that an integrally closed domain is divisorial if and only if it is a Prüfer domain with certain finiteness properties [19,Theorem 5.1].…”

Section: Introductionmentioning

confidence: 99%

Note on the divisoriality of domains of the form $k[[X^{p}, X^{q}]]$, $k[X^{p}, X^{q}]$, $k[[X^{p}, X^{q}, X^{r}]]$, and $k[X^{p}, X^{q}, X^{r}]$

Mimouni¹

2016

Turk J Math

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

Note on the divisoriality of domains of the form $k[[X^{p}, X^{q}]]$, $k[X^{p}, X^{q}]$, $k[[X^{p}, X^{q}, X^{r}]]$, and $k[X^{p}, X^{q}, X^{r}]$

Mimouni¹

2016

Turk J Math

View full text Add to dashboard Cite

show abstract

“…Divisoriality is a local property, in the sense that a domain R is divisorial if and only if it is h-local (for the definition see after Theorem 3.2) and every localization of R at a maximal ideal is divisorial; the necessity of this fact was proved by Matlis and Heinzer independently, while the sufficiency was proved by Bazzoni-Salce in [4]. A complete characterization of divisorial domains was obtained only very recently by Bazzoni [3].…”

Section: Torsionless and Divisorial Domainsmentioning

confidence: 93%

Warfield Domains: Module Theory from Linear Algebra to Commutative Algebra Through Abelian Groups

Salce

2002

Milan Journal of Mathematics

View full text Add to dashboard Cite

In this survey paper we illustrate the generalizations to modules over integral domains of the basic result of linear algebra stating that every finite dimensional vector space is canonically isomorphic to its bidual. These generalizations culminate in the discovery of Warfield domains. We present the characterization of Warfield domains in terms of properties of their overrings, as well as their intrinsic characterizations in the classical cases of Noetherian and integrally closed domains, and in the general case. Finally, the development of the results on direct sum decompositions of torsionless modules from the classical case of Dedekind domains up to Warfield domains is sketched; this subject was originated by the 1912 result of Steinitz on finite direct sums of ideals in algebraic number rings.

show abstract

“…On the other hand, if any overring of a G-Dedekind domain R is still a G-Dedekind domain, then R must be totally divisorial and further, totally reflexive since it is a Noetherian domain. Therefore, by the above two theorems or [3,Theorem 7.3], R must be a Warfield domain and all the ideals of R are 2-generated (the definition of a Warfield domain can be seen in [19]). Next, we will prove that the G-Dedekind domain Q + X 2 Q[X] is a Warfield domain.…”

Section: Overrings Of G-dedekind Domainsmentioning

confidence: 99%

On Overrings of Gorenstein Dedekind Domains

Hu¹,

Wang²,

Xu³

et al. 2013

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

Abstract. In this paper, we mainly discuss Gorenstein Dedekind domains (G-Dedekind domains for short) and their overrings. Let R be a one-dimensional Noetherian domain with quotient field K and integral closure T . Then it is proved that R is a G-Dedekind domain if and only if for any prime ideal P of R which contains (R : K T ), P is Gorenstein projective. We also give not only an example to show that G-Dedekind domains are not necessarily Noetherian Warfield domains, but also a definition for a special kind of domain: a 2-DVR. As an application, we prove that a Noetherian domain R is a Warfield domain if and only if for any maximal ideal M of R, R M is a 2-DVR.

show abstract

Warfield Domains

Cited by 57 publications

References 0 publications

Note on the divisoriality of domains of the form $k[[X^{p}, X^{q}]]$, $k[X^{p}, X^{q}]$, $k[[X^{p}, X^{q}, X^{r}]]$, and $k[X^{p}, X^{q}, X^{r}]$

Note on the divisoriality of domains of the form $k[[X^{p}, X^{q}]]$, $k[X^{p}, X^{q}]$, $k[[X^{p}, X^{q}, X^{r}]]$, and $k[X^{p}, X^{q}, X^{r}]$

Warfield Domains: Module Theory from Linear Algebra to Commutative Algebra Through Abelian Groups

On Overrings of Gorenstein Dedekind Domains

Contact Info

Product

Resources

About