On strongly primary monoids and domains

Geroldinger, Alfred; Roitman, Moshe

doi:10.1080/00927872.2020.1755678

Cited by 14 publications

(9 citation statements)

References 30 publications

(61 reference statements)

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“…By Corollary 3.2.1 , every strongly primary stable domain is Mori. This is not true for general strongly primary domains [ 26 , Section 3] and it is in strong contrast to other classes of strongly primary monoids [ 19 , Theorem 3.3]. By [ 15 , Example 5.17], there exists a stable two-dimensional Archimedean local integral domain.…”

Section: Stable Domainsmentioning

confidence: 99%

“…Thus all assumptions of Proposition 4.3 are satisfied. Let Since is a one-dimensional local Mori domain, it is strongly primary and hence locally tame by [ 26 , Theorem 3.9]. Thus its catenary degree is finite by [ 19 , Theorem 4.1].…”

Section: Arithmetic Of Stable Orders In Dedekind Domainsmentioning

confidence: 99%

“…Thus its catenary degree is finite by [ 19 , Theorem 4.1]. If then by [ 26 , Theorem 3.7]. Suppose that Then is finitely primary by [ 21 , Proposition 2.10.7], whence the claim on the elasticity follows from (5.3) and (5.4) .…”

Section: Arithmetic Of Stable Orders In Dedekind Domainsmentioning

confidence: 99%

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On the arithmetic of stable domains

Bashir

Geroldinger

Reinhart

2021

Communications in Algebra

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A commutative ring R is stable if every non-zero ideal I of R is projective over its ring of endomorphisms. Motivated by a paper of Bass in the 1960s, stable rings have received wide attention in the literature ever since then. Much is known on the algebraic structure of stable rings and on the relationship of stability with other algebraic properties such as divisoriality and the 2-generator property. In the present paper, we study the arithmetic of stable integral domains, with a focus on arithmetic properties of semigroups of ideals of stable orders in Dedekind domains.

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Section: Stable Domainsmentioning

confidence: 99%

Section: Arithmetic Of Stable Orders In Dedekind Domainsmentioning

confidence: 99%

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On the arithmetic of stable domains

Bashir

Geroldinger

Reinhart

2021

Communications in Algebra

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“…We continue with a few more highlights to indicate the importance of transfer Krull monoids. In [21] it is proved that a strongly primary monoid is transfer Krull if and only if it is half-factorial. Moreover, a length-factorial monoid is transfer Krull if and only if it is Krull ( [24]).…”

Section: Introductionmentioning

confidence: 99%

On transfer Krull monoids

Bashir,

Reinhart

2021

Preprint

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Let H be a cancellative commutative monoid, let A(H) be the set of atoms of H and let H be the root closure of H. Then H is called transfer Krull if there exists a transfer homomorphism from H into a Krull monoid. It is well known that both half-factorial monoids and Krull monoids are transfer Krull monoids. In spite of many examples and counter examples of transfer Krull monoids (that are neither Krull nor half-factorial), transfer Krull monoids have not been studied systematically (so far) as objects on their own. The main goal of the present paper is to attempt the first in-depth study of transfer Krull monoids. We investigate how the root closure of a monoid can affect the transfer Krull property and under what circumstances transfer Krull monoids have to be half-factorial or Krull. In particular, we show that if H is a DVM, then H is transfer Krull if and only if H ⊆ H is inert. Moreover, we prove that if H is factorial, then H is transfer Krull if and only if AWe also show that if H is half-factorial, then H is transfer Krull if and only if A(H) ⊆ A( H). Finally, we point out that characterizing the transfer Krull property is more intricate for monoids whose root closure is Krull. This is done by providing a series of counterexamples involving reduced affine monoids.

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“…Strongly primary monoids are the objects of study in the present paper, and we discuss them in Subsection 2.4. If H is primary, then it follows from [22, Proposition 1] and[33, Lemma 2.5] that…”

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confidence: 99%

On strongly primary monoids, with a focus on Puiseux monoids

Geroldinger,

Gotti,

Tringali

2019

Preprint

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Primary and strongly primary monoids and domains play a central role in the ideal and factorization theory of commutative monoids and domains. Among others, it is known that primary monoids satisfying the ascending chain condition on divisorial ideals (e.g., numerical monoids) are strongly primary; and the multiplicative monoid of non-zero elements of a one-dimensional local domain is primary and it is strongly primary if the domain is noetherian. In the present paper, we focus on the study of additive submonoids of the non-negative rationals, called Puiseux monoids. It is easy to see that Puiseux monoids are primary monoids, and we provide conditions ensuring that they are strongly primary. Then we study local and global tameness of strongly primary Puiseux monoids; most notably, we establish an algebraic characterization of when a Puiseux monoid is globally tame. Moreover, we obtain a result on the structure of sets of lengths of all locally tame strongly primary monoids.

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On strongly primary monoids and domains

Cited by 14 publications

References 30 publications

On the arithmetic of stable domains

On the arithmetic of stable domains

On transfer Krull monoids

On strongly primary monoids, with a focus on Puiseux monoids

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