Hereditary atomicity in integral domains

Coykendall, Jim; Gotti, Felix; Hasenauer, Richard Erwin

doi:10.1016/j.jpaa.2022.107249

Cited by 2 publications

References 52 publications

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On the atomic structure of torsion-free monoids

Gotti,

Vulakh

2023

Semigroup Forum

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Let M be a cancellative and commutative (additive) monoid. The monoid M is atomic if every non-invertible element can be written as a sum of irreducible elements, which are also called atoms. Also, M satisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. In the first part of this paper, we characterize torsion-free monoids that satisfy the ACCP as those torsion-free monoids whose submonoids are all atomic. A submonoid of the nonnegative cone of a totally ordered abelian group is often called a positive monoid. Every positive monoid is clearly torsion-free. In the second part of this paper, we study the atomic structure of certain classes of positive monoids.

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On the atomic structure of torsion-free monoids

Gotti,

Vulakh

2023

Semigroup Forum

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Divisibility in rings of integer-valued polynomials

Gotti¹,

Li²

2021

Preprint

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In this paper, we address various aspects of divisibility by irreducibles in rings consisting of integer-valued polynomials. An integral domain is called atomic if every nonzero nonunit factors into irreducibles. Atomic domains that do not satisfy the ascending chain condition on principal ideals (ACCP) have proved to be elusive, and not many of them have been found since the first one was constructed by A. Grams in 1974. Here we exhibit the first class of atomic rings of integer-valued polynomials without the ACCP. An integral domain is called a finite factorization domain (FFD) if it is simultaneously atomic and an idf-domain (i.e., every nonzero element is divisible by only finitely many irreducibles up to associates). We prove that a ring is an FFD if and only if its ring of integer-valued polynomials is an FFD. In addition, we show that neither being atomic nor being an idf-domain transfer, in general, from an integral domain to its ring of integer-valued polynomials. In the same class of rings of integer-valued polynomials, we consider further properties that are defined in terms of divisibility by irreducibles, including being Cohen-Kaplansky and being Furstenberg.

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Hereditary atomicity in integral domains

Cited by 2 publications

References 52 publications

On the atomic structure of torsion-free monoids

On the atomic structure of torsion-free monoids

Divisibility in rings of integer-valued polynomials

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