Radical factorization in finitary ideal systems

Olberding, Bruce; Reinhart, Andreas

doi:10.1080/00927872.2019.1640237

Cited by 4 publications

(2 citation statements)

References 21 publications

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“…In this section, we study, for a finitary ideal system r of a cancellative monoid H , algebraic and arithmetic properties of the semigroup of r-ideals and of the semigroup of r-invertible r-ideals . A focus is on the question when these monoids of r -ideals are half-factorial (other arithmetical properties of such as radical factoriality, were recently studied in [ 48 ]). In Section 5 , we apply these results to monoids of divisorial ideals and to monoids of usual ring ideals of Mori domains.…”

Section: Monoids Of Ideals and Half-factorialitymentioning

confidence: 99%

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: Monoids Of Ideals and Half-factorialitymentioning

confidence: 99%

On the arithmetic of stable domains

Bashir

Geroldinger

Reinhart

2021

Communications in Algebra

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“…In this section we study, for a finitary ideal system r of a cancellative monoid H, algebraic and arithmetic properties of the semigroup I r (H) of r-ideals and of the semigroup I * r (R) of r-invertible rideals. A focus is on the question when these monoids of r-ideals are half-factorial (other arithmetical properties of I * r (H), such as radical factoriality, were recently studied in [48]). In Section 5, we apply these results to monoids of divisorial ideals and to monoids of usual ring ideals of Mori domains.…”

Section: Monoids Of Ideals and Half-factorialitymentioning

confidence: 99%

On the arithmetic of stable domains

Bashir¹,

Geroldinger²,

Reinhart³

2020

Preprint

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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. Initiated 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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Ideal factorization in strongly discrete independent rings of Krull type

Chang

Choi

2021

J. Algebra Appl.

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Let [Formula: see text] be an integral domain and [Formula: see text] be the so-called [Formula: see text]-operation on [Formula: see text]. In this paper, we define the notion of [Formula: see text]-ZPUI domains which is a natural generalization of ZPUI domains introduced by Olberding in 2000. We say that [Formula: see text] is a [Formula: see text]-ZPUI domain if every nonzero proper [Formula: see text]-ideal [Formula: see text] of [Formula: see text] can be written as [Formula: see text] for some [Formula: see text]-invertible ideal [Formula: see text] of [Formula: see text] and [Formula: see text] is a nonempty collection of pairwise [Formula: see text]-comaximal prime [Formula: see text]-ideals of [Formula: see text]. Then, among other things, we show that [Formula: see text] is a [Formula: see text]-ZPUI domain if and only if the polynomial ring [Formula: see text] is a [Formula: see text]-ZPUI domain, if and only if [Formula: see text] is a strongly discrete independent ring of Krull type. We construct three types of new [Formula: see text]-ZPUI domains from an old one by [Formula: see text]-construction, pullback, and [Formula: see text]-domains. We also show that given an abelian group [Formula: see text], there is a ZPUI domain with ideal class group [Formula: see text] but not a Dedekind domain. Finally, we introduce and study the notion of [Formula: see text]-ISP domains as a generalization of [Formula: see text]-ZPUI domains.

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Radical factorization in finitary ideal systems

Cited by 4 publications

References 21 publications

On the arithmetic of stable domains

On the arithmetic of stable domains

On the arithmetic of stable domains

Ideal factorization in strongly discrete independent rings of Krull type

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