A lower bound for the number of intermediary rings

Jaballah, Ali

doi:10.1080/00927879908826495

Cited by 27 publications

(8 citation statements)

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“…An integral domain is said to be FO (or FC) if the corresponding condition is satisfied for the extension R ⊆ K, where K is the field of fractions of R. Several characterizations of extensions R ⊆ K satisfying these conditions have been established by Gilmer in [7]. Several related results can be found in [9], [11], [3] and [2]. We investigate in this paper the realization of these two conditions in the more general setting of extensions of integral domains, where the upper ring S is not necessarily the field of quotients of the ring R. We establish in Theorem 2.2 and in Theorem 3.2 of this paper several characterizations of these extensions.…”

Section: Introductionmentioning

confidence: 90%

Ring extensions with some finiteness conditions on the set of intermediate rings

Jaballah

2010

Czech Math J

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Section: Introductionmentioning

confidence: 90%

Ring extensions with some finiteness conditions on the set of intermediate rings

Jaballah

2010

Czech Math J

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“…We begin by some observations in the case [D, Introduction]. Therefore, there is a prime ideal of R, say Q i such that Recall the following result due to A. Jaballah [J,Lemma 3.2]. We label it as Lemma 5 for the sake of reference.…”

Section: The Case R Is Integrally Closed In Smentioning

confidence: 96%

Some finiteness chain conditions on the set of intermediate rings

Ayache

2010

Journal of Algebra

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Let R ⊂ S be an extension of integral domains and [R, S] be the set of intermediate rings between R and S. We say that [R, S] satisfies the finite chain condition (FCC) if every chain of distinct intermediate rings between R and S is finite. Our main purpose is to determine necessary and sufficient conditions so that [R, S]satisfies FCC. We also investigate the relationship between FCC and other finiteness conditions. Several satisfactory results are settled, but special attention is focused on the case where R is a Prüfer ring or a Noetherian ring.

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“…That is Several papers have provided approximations for the cardinality |[R, S]| in the context of normal pairs with finite supports [3][4][5]12]. Recently, [6] has presented an effective algorithm that enables us to compute the exact value of |[R, S]|.…”

Section: Let a = Supp(s/r) ∪ {O} Ordered By Inclusion Where O Is Thementioning

confidence: 97%

The set of indeterminate rings of a normal pair as a partially ordered set

Ayache

2010

Ricerche mat.

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Let R ⊂ S be an extension of integral domains and let [R, S] be the set of intermediate rings between R and S ordered by inclusion. If (R, S) is normal pair and [R, S] is finite, we do prove that there exists a semi-local Prüfer ring T with quotient field K such that [R, S] ∼ = [T, K ] (as a partially ordered set). Consequently, any problem relative to the finiteness conditions in [R, S] can be investigated in the particular case where R is a semi-local Prüfer ring with quotient field S.

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A lower bound for the number of intermediary rings

Cited by 27 publications

References 1 publication

Ring extensions with some finiteness conditions on the set of intermediate rings

Ring extensions with some finiteness conditions on the set of intermediate rings

Some finiteness chain conditions on the set of intermediate rings

The set of indeterminate rings of a normal pair as a partially ordered set

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