Prüfer ∗ -multiplication domains and torsion theories

García, Josefa M.; Jara, Pascual; Santos, Eva

doi:10.1080/00927879908826493

Cited by 13 publications

(11 citation statements)

References 7 publications

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“…also J.M. García, P. Jara and E. Santos [25]). We give a characterization theorem of these domains in terms of Kronecker function rings and Nagata rings associated naturally to the given semistar operation, generalizing previous results by J. Arnold and J.…”

Section: Introductionmentioning

confidence: 99%

“…This class contains as examples Prüfer domains, Krull domains and PvMD, but also integral domains, that are not integrally closed, having although an appropriate overring which is Prüfer star multiplicative domain (cf. [36], [39] and [25]). An explicit example of a non integrally closed Prüfer semistar multiplication domain is given in Example (3.10) (recall that a Prüfer star multiplication domain is always integrally closed).…”

Section: Introductionmentioning

confidence: 99%

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Prüfer ⋆-Multiplication Domains and Semistar Operations

2003

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Starting from the notion of semistar operation, introduced in 1994 by Okabe and Matsuda [49], which generalizes the classical concept of star operation (cf. Gilmer's book [27]) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer, P. Lorenzen and P. Jaffard (cf. Halter-Koch's book [32]), in this paper we outline a general approach to the theory of Prüfer ⋆-multiplication domains (or P⋆MDs), where ⋆ is a semistar operation. This approach leads to relax the classical restriction on the base domain, which is not necessarily integrally closed in the semistar case, and to determine a semistar invariant character for this important class of multiplicative domains (cf. also J.M. García, P. Jara and E. Santos [25]). We give a characterization theorem of these domains in terms of Kronecker function rings and Nagata rings associated naturally to the given semistar operation, generalizing previous results by J. Arnold and J. Brewer [10] and B.G. Kang [39]. We prove a characterization of a P⋆MD, when ⋆ is a semistar operation, in terms of polynomials (by using the classical characterization of Prüfer domains, in terms of polynomials given by . We also deal with the preservation of the P⋆MD property by "ascent" and "descent" in case of field extensions. In this context, we generalize to the P⋆MD case some classical results concerning Prüfer domains and PvMDs. In particular, we reobtain as a particular case a result due to H. Prüfer [51] and W. Krull [41] (cf. also F. Lucius ). Finally, we develop several examples and applications when ⋆ is a (semi)star given explicitly (e.g. we consider the case of the "standard" v-, t-, b-, w-operations or the case of semistar operations associated to appropriate families of overrings).

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Prüfer ⋆-Multiplication Domains and Semistar Operations

2003

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“…A number of authors (Anderson, 1988;Anderson et al, 1989a,b;Anderson and Cook, 2000;Anderson and Clarke, 2005;Fontana and Huckaba, 2000;Garcia et al, 1999;Halter-Koch, 2000, 2001 …”

Section: Star-operationsmentioning

confidence: 98%

When thev-Operation Distributes over Intersections

Anderson

Clarke

2006

Communications in Algebra

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“…Note also that, if D is * -Noetherian, then each ideal of D is * -finite, but the converse is false in general (cf. [3, page 29] and[20, Example 18]).…”

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confidence: 99%