AP-domains and unique factorization

Coykendall, Jim; Zafrullah, Muhammad

doi:10.1016/j.jpaa.2003.10.036

Cited by 13 publications

(20 citation statements)

References 6 publications

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“…In the following example from [25], we will produce a nonatomic domain that has the property that any two equal irreducible factorizations have the same length. Example 2.3.…”

Section: Some Preliminary Results and Definitionsmentioning

confidence: 99%

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Extensions of Half-Factorial Domains

Coykendall¹

2005

Lecture Notes in Pure and Applied Mathematics

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A half-factorial domain (HFD) is an atomic domain, R, with the property that if one has the irreducible factorizations in RHalf-factorial domains were implicitly studied in the case of rings of algebraic integers in a 1960 paper of L. Carlitz [11] and were subsequently abstracted and studied by A. Zaks.Since their inception as a generalization of the classical notion of unique factorization domains, half-factorial domains have been the subject of much interest in commutative algebra. This paper will give a survey of some recent advances in the study of half-factorial domains with the emphasis on advances in the understanding of ring extension behavior of half-factorial domains.

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“…In the following example from [25], we will produce a nonatomic domain that has the property that any two equal irreducible factorizations have the same length. Example 2.3.…”

Section: Some Preliminary Results and Definitionsmentioning

confidence: 99%

“…The example is a domain with a unique nonprime irreducible element (up to associates) and was constructed in [25]. In this domain, the uniqueness of the irreducible assures that any two irreducible factorizations are of the same length (in fact, unique).…”

Section: Some Preliminary Results and Definitionsmentioning

confidence: 99%

Extensions of Half-Factorial Domains

Coykendall¹

2005

Lecture Notes in Pure and Applied Mathematics

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“…The inductive construction of the domain in the proof of this theorem bears some resemblance to Artin's construction of the algebraic closure of a field in [11]. The examples in [7], [13], and in Proposition 2.4 of [12] are also constructed inductively.…”

Section: Introductionmentioning

confidence: 92%

Polynomial extensions of IDF-domains and of IDPF-domains

Malcolmson

Okoh

2008

Proc. Amer. Math. Soc.

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Abstract. An integral domain is IDF if every non-zero element has only finitely many non-associate irreducible divisors. We investigate when R IDF implies that the ring of polynomials R[T ] is IDF. This is true when R is Noetherian and integrally closed, in particular when R is the coordinate ring of a non-singular variety. Some coordinate rings R of singular varieties also give R[T ] IDF. Analogous results for the related concept of IDPF are also given. The main result on IDF in this paper states that every countable domain embeds in another countable domain R such that R has no irreducible elements, hence vacuously IDF, and the polynomial ring R[T ] is not IDF. This resolves an open question. It is also shown that some subrings R of the ring of Gaussian integers known to be IDPF also have the property that R[T ] is not IDPF.

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“…The first example shows that every integral domain is contained in an integral domain (not a field) with no atoms. This type of integral domain has been studied extensively [3] We also note that the technique of Theorem 2.7 was used in [5] to construct an integral domain where all the finite atom factorizations are unique, but the atoms need not be prime. Proof.…”

Section: Applicationsmentioning

confidence: 99%

“…He started with R = F [Z, { Roitman later showed that the power series ring A[[X]] also need not be atomic when A is an atomic domain [11]. This paper begins by revisiting results concerning Roitman's construction that are found in [1], [4], and [5]. Anderson and Anderson explored the factorization properties of an element r in R, R[X, r/X], and R[X] in [1].…”

Section: Introductionmentioning

confidence: 99%

Multiplicative Subsets of Atoms

Rand¹

2015

International Electronic Journal of Algebra

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Abstract. A reduced, cancellative, torsion-free, commutative monoid M can be embedded in an integral domain R, where the atoms (irreducible elements) of M correspond to a subset of the atoms of R. This fact was used by J.Coykendall and B. Mammenga to show that for any reduced, cancellative, torsion-free, commutative, atomic monoid M , there exists an integral domain R with atomic factorization structure isomorphic to M . More generally, we show that any "nice" subset of atoms of R can be realized as the set of atoms of an integral domain T that contains R. We will also give several applications of this result.Mathematics Subject Classification (2010): 20M14, 20M15, 13G05

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AP-domains and unique factorization

Cited by 13 publications

References 6 publications

Extensions of Half-Factorial Domains

Extensions of Half-Factorial Domains

Polynomial extensions of IDF-domains and of IDPF-domains

Multiplicative Subsets of Atoms

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