On the prime spectrum of a mori domain

Barucci, Valentina; Houston, Evan

doi:10.1080/00927879608825773

Cited by 20 publications

(8 citation statements)

References 16 publications

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“…Accordingly, in a recent paper [3], Barucci and Houston ask if R & R Ã satisfies LO, where R is a (commutative integral) domain with complete integral closure R Ã . More precisely, in [3, page 3601], Barucci and Houston raise this question in case R is Archimedean, as they obtain a negative answer for any non-Archimedean domain R. (Recall from [15] that a domain R is said to be Archimedean in case Rr n 0 for each nonunit r P R. Natural examples of Archimedean domains include Noetherian domains and domains of Krull dimension 1, as a consequence of [ [12]) and to lying-over for certain prime ideals (the PF-prime ideals, in the sense of [17]).…”

Section: Introductionmentioning

confidence: 99%

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Prime ideals surviving in complete integral closures

Dobbs

1997

Archiv der Mathematik

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Section: Introductionmentioning

confidence: 99%

“…In connection with the question of Barucci and Houston [3] which was mentioned in the Introduction, it is useful to note that the proof of Theorem 2.2 may be modified to establish the following result. …”

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confidence: 99%

Prime ideals surviving in complete integral closures

Dobbs

1997

Archiv der Mathematik

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“…Using the idea of conductors, we give a simplified proof to show that the complete integral closure of a semi normal Mori domain having a non zero pseudo radical is a Krull domain. On the other hand, if M ∈ S(D), using [7], D = (M :…”

Section: Definitionmentioning

confidence: 99%

Some results on a conjecture regarding Mori domain

Gebru

2011

Math. J. Ibaraki Univ.

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Based on the famous Mori-Nagata Theorem: The integral closure of a noetherian domain is a Krull domain, similar assertion was conjectured for Mori domain as follows: The complete integral closure of a Mori domain is a Krull domain. The conjecture is positive for a noetherian domain, Krull domain, a semi normal Mori domain [6] and Mori domains for which (D : D) = 0. In general, as M. Roitman has noted [26], the conjecture is not true. In this paper, an attempt is being made, among other things, to prove that the conjecture is true for a one dimensional Mori domain and for a finite dimensional AV-Mori domain. On the other hand, using the idea of conductor ideals, a simplified proof is given that the conjecture is true for semi normal Mori domains with nonzero pseudo radical.

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“…Any divisorial ideal is a w−ideal. Now, R is said to be a Mori domain if it satisfies the ascending chain condition on divisorial ideals [5,6,8,25] and a strong Mori domain if it satisfies the ascending chain condition on w−ideals [20,35]. Trivially, a Noetherian domain is strong Mori and a strong Mori domain is Mori.…”

Section: Noetherian-like Settingsmentioning

confidence: 99%

Class semigroups of integral domains

Kabbaj

Mimouni

2003

Journal of Algebra

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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 (Lipman and Sally-Vasconcelos) stability conditions; a main feature is that the Clifford property forces t−locally Noetherian domains to be one-dimensional Noetherian domains. Section 5 studies the transfer of the Clifford and Boolean properties to various pullback constructions. Our results lead to new families of integral domains with Clifford or Boolean class semigroup, moving therefore beyond the contexts of integrally closed domains or Noetherian domains.

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On the prime spectrum of a mori domain

Cited by 20 publications

References 16 publications

Prime ideals surviving in complete integral closures

Prime ideals surviving in complete integral closures

Some results on a conjecture regarding Mori domain

Class semigroups of integral domains

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