On semigroup algebras with rational exponents

Gotti, Felix

doi:10.1080/00927872.2021.1949018

Cited by 21 publications

(10 citation statements)

References 35 publications

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“…For any p ∈ P, the positive monoid M p = 1/p n | n ∈ N 0 is antimatter and, therefore, an IDF-domain. However, it was proved in [29,Example 5.5] that the monoid domain Q[M p ] is not an IDF-domain. As being antimatter violates too drastically the Furstenberg condition, here we exhibit an IDF-monoid M with any prescribed size |A (M )| such that the monoid domain Q[M ] is not an IDF-domain.…”

Section: Polynomial Rings and Monoid Rings As Proved By Malcolmson An...mentioning

confidence: 99%

Integral domains and the IDF property

Gotti¹,

Zafrullah²

2022

Preprint

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An integral domain D is called an irreducible-divisor-finite domain (IDF-domain) if every nonzero element of D has finitely many irreducible divisors up to associates. The study of IDF-domains dates back to the seventies. In this paper, we investigate various aspects of the IDF property. In 2009, P. Malcolmson and F. Okoh proved that the IDF property does not ascend from integral domains to their corresponding polynomial rings, answering a question posed by D. D. Anderson, D. F. Anderson, and the second author two decades before. Here we prove that the IDF property ascends in the class of PSP-domains, generalizing the known result (also by Malcolmson and Okoh) that the IDF property ascends in the class of GCD-domains. We put special emphasis on IDF-domains where every nonunit is divisible by an irreducible, which we call TIDF-domains, and we also consider PIDF-domains, which form a special class of IDF-domains introduced by Malcolmson and Okoh in 2006. We investigate both the TIDF and the PIDF properties under taking polynomial rings and localizations. We also delve into their behavior under monoid domain and D + M constructions.

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Section: Polynomial Rings and Monoid Rings As Proved By Malcolmson An...mentioning

confidence: 99%

Integral domains and the IDF property

Gotti¹,

Zafrullah²

2022

Preprint

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“…The primary purpose of this section is to construct a new class of atomic monoid algebras that do not satisfy the ACCP. In order to do so, we consider monoid domains with coefficients in a field and exponents in the nonnegative ray of R. Several classes of atomic monoid domains with coefficients in a field and exponents in the nonnegative ray of Q were recently considered by the first author in [13]. However, every atomic monoid domain considered in the mentioned paper satisfies the ACCP.…”

Section: Atomic Semigroup Rings Without the Accpmentioning

confidence: 99%

Atomic semigroup rings and the ascending chain condition on principal ideals

Gotti¹,

Li²

2021

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An integral domain is called atomic if every nonzero nonunit element factors into irreducibles. On the other hand, an integral domain is said to satisfy the ascending chain condition on principal ideals (ACCP) if every ascending chain of principal ideals terminates. It was asserted by Cohn back in the sixties that every atomic domain satisfies the ACCP, but such an assertion was refuted by Grams in the seventies with an explicit construction of a neat example. Still, atomic domains without the ACCP are notoriously elusive, and just a few classes have been found since Grams' first construction. In the first part of this paper, we generalize Grams' construction to provide new classes of atomic domains without the ACCP. In the second part of this paper, we construct what seems to be the first atomic semigroup ring without the ACCP in the existing literature.

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“…As a result, F p [M ] is a hereditarily atomic domain. Several aspects of the atomicity of F [M ] for additive submonoids M of Q ≥0 (and any field F ) were recently studied by the second author in [26].…”

Section: Polynomial-like Ringsmentioning

confidence: 99%

Hereditary atomicity in integral domains

Coykendall¹,

Gotti²,

Hasenauer³

2021

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If every subring of an integral domain is atomic, we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind domains in terms of atomic overrings. Then we use this characterization to determine the fields that are hereditarily atomic. On the other hand, we investigate hereditary atomicity in the context of rings of polynomials and rings of Laurent polynomials, characterizing the fields and rings whose rings of polynomials and rings of Laurent polynomials, respectively, are hereditarily atomic. As a result, we obtain two classes of hereditarily atomic domains that cannot be embedded into any hereditarily atomic field. By contrast, we show that rings of power series are never hereditarily atomic. Finally, we make some progress on the still open question of whether every subring of a hereditarily atomic domain satisfies ACCP.

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On semigroup algebras with rational exponents

Cited by 21 publications

References 35 publications

Integral domains and the IDF property

Integral domains and the IDF property

Atomic semigroup rings and the ascending chain condition on principal ideals

Hereditary atomicity in integral domains

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