Finitary monoids

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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“…Proof. Since R is a seminormal G-domain, ðR : b RÞ 6 ¼ f0g by [22,Proposition 4.8]. Thus the claim follows from Theorem 3.7. w Example 3.9.…”

Section: Stable Domainsmentioning

confidence: 79%

On the arithmetic of stable domains

Bashir

Geroldinger

Reinhart

2021

Communications in Algebra

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“…We have that S P is an overring of R P , and thus S P is a G-domain, as well. Moreover, S P is seminormal, and so it follows by [15,Proposition 4.8.2] that (S P : R P ) ⊇ (S P : S P ) = 0. We conclude that (R P : R P ) ⊇ (R P : S P )(S P : R P ) = 0, and the claim is thus proved.…”

Section: Finite Molecularization Domainsmentioning

confidence: 99%

On finite molecularization domains

Hetzel¹,

Lawson²,

Reinhart³

2021

Preprint

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In this paper, we advance an ideal-theoretic analogue of a "finite factorization domain" (FFD), giving such a domain the moniker "finite molecularization domain" (FMD). We characterize FMD's as those factorable domains (termed "molecular domains" in the paper) for which every nonzero ideal is divisible by only finitely many nonfactorable ideals (termed "molecules" in the paper) and the monoid of nonzero ideals of the domain is unit-cancellative, in the language of Fan, Geroldinger, Kainrath, and Tringali. We develop a number of connections, particularly at the local level, amongst the concepts of "FMD", "FFD", and the "finite superideal domains" (FSD's) of Hetzel and Lawson. Characterizations of when k[X 2 , X 3 ], where k is a field, and the classical D + M construction are FMD's are provided. We also demonstrate that if R is a Dedekind domain with the finite norm property, then R[X] is an FMD.

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“…Let thus for some and all Let Since H is primary, we have for some so for all Thus, for all Since H is root-closed by item 1., we obtain that for all Thus, so Since by Lemma 2.3 , we infer that that is, so Cf. [ 19 , Proposition 4.8]. □…”

Section: Background On Primary Monoids and Domainsmentioning

confidence: 99%

On strongly primary monoids and domains

Geroldinger

Roitman

2020

Communications in Algebra

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A commutative integral domain is primary if and only if it is one-dimensional and local. A domain is strongly primary if and only if it is local and each nonzero principal ideal contains a power of the maximal ideal. Hence, one-dimensional local Mori domains are strongly primary. We prove among other results that if R is a domain such that the conductor ðR :RÞ vanishes, then KðRÞ is finite; that is, there exists a positive integer k such that each nonzero nonunit of R is a product of at most k irreducible elements. Using this result, we obtain that every strongly primary domain is locally tame, and that a domain R is globally tame if and only if KðRÞ ¼ 1: In particular, we answer Problem 38 of the open problem list by Cahen et al. in the affirmative. Many of our results are formulated for monoids.

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Finitary monoids

Cited by 15 publications

References 0 publications

On the arithmetic of stable domains

On the arithmetic of stable domains

On finite molecularization domains

On strongly primary monoids and domains

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