Minimal Overrings of an Integrally Closed Domain

Ayache, Ahmed

doi:10.1081/agb-120024849

Cited by 22 publications

(15 citation statements)

References 9 publications

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“…If R is a domain, the classification in [10,Theorem 2.7] shows that any such T must be R-algebra isomorphic to an overring of R. We will see that the same holds true more generally, namely, if tq(R) is von Neumann regular. The main results in this section generalize the work of Ayache [2] who showed, among other things, that if R is an integrally closed domain but not a field, then each minimal overring of R is the Kaplansky transform of a maximal ideal that satisfies certain properties. In [21], a generalized notion of the Kaplansky transform was introduced and developed.…”

Section: The Integrally Closed Casementioning

confidence: 68%

“…If any of (1), (2) or (3) Suppose next that (3) holds. As q / ∈ R, we have, a fortiori, that q / ∈ M. Consider the nonzero element y := q + M ∈ T /M.…”

Section: Moreover the Above Listing Is A Classification In The Follmentioning

confidence: 97%

“…The characterization in Theorem 3.1 generalizes a result of Ayache [2, Theorem 1.2] on quasilocal integrally closed domains that are not fields. As in [2], the characterization in Theorem 3.1 (as well as that in Theorem 3.7) depends on the notion of a divided prime ideal, in the sense of [3,4]. It also depends on the notion of the Kaplansky transform of a maximal ideal.…”

Section: Introductionmentioning

confidence: 95%

“…Theorem 3.7 generalizes this result in two ways: by assuming only that tq(R) is von Neumann regular; and by characterizing the minimal ring extensions T of R in which R is integrally closed. Insofar as possible, our path to Theorem 3.7 is patterned after the approach in [2], although we must once again use the generalized Kaplansky transform from [21]. As was the case in Theorem 3.1, it turns out in Theorem 3.7 that any such T must be an overring of R and, up to R-algebra isomorphism, the Kaplansky transform of a uniquely determined maximal ideal of R.…”

Section: Introductionmentioning

confidence: 97%

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A classification of the minimal ring extensions of certain commutative rings

Dobbs

Shapiro

2007

Journal of Algebra

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All rings considered are commutative with identity and all ring extensions are unital. Let R be a ring with total quotient ring T . The integral minimal ring extensions of R are catalogued via generator-and-relations. If T is von Neumann regular and no maximal ideal of R is a minimal prime ideal of R, the minimal ring extensions of R are classified, up to R-algebra isomorphism, as the minimal overrings (within T ) of R and, for maximal ideals M of R, the idealizations R(+)R/M and the direct products R × R/M. If T is von Neumann regular, the minimal ring extensions of R in which R is integrally closed are characterized as certain overrings, up to R-algebra isomorphism, in terms of Kaplansky transforms and divided prime ideals, generalizing work of Ayache on integrally closed domains; no restriction on T is needed if R is quasilocal. One application generalizes a recently announced result of Picavet and Picavet-L'Hermitte on the minimal overrings of a local Noetherian ring. Examples are given to indicate sharpness of the results.

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Section: The Integrally Closed Casementioning

confidence: 68%

“…If any of (1), (2) or (3) Suppose next that (3) holds. As q / ∈ R, we have, a fortiori, that q / ∈ M. Consider the nonzero element y := q + M ∈ T /M.…”

Section: Moreover the Above Listing Is A Classification In The Follmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 97%

See 2 more Smart Citations

A classification of the minimal ring extensions of certain commutative rings

Dobbs

Shapiro

2007

Journal of Algebra

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“…Set X 1 = A\{m 0 }, then the restriction 1 of ϕ is also an order isomorphism. It follows that the composite function ϕ 1 = ϕ 1 • ψ 1 from Spec(R 1 ) to X 1 is an order isomorphism.…”

Section: Lemma 3 Let a Be A Tree With A Unique Minimal Element O And mentioning

confidence: 98%

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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About minimal morphisms

Picavet

Picavet-L’Hermitte

Multiplicative Ideal Theory in Commutative Algebra

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Minimal Overrings of an Integrally Closed Domain

Cited by 22 publications

References 9 publications

A classification of the minimal ring extensions of certain commutative rings

A classification of the minimal ring extensions of certain commutative rings

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

About minimal morphisms

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