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Divisibility and a weak ascending chain condition on principal ideals
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abstract
An integral domain $R$ is atomic if each nonzero nonunit of $R$ factors into irreducibles. In addition, an integral domain $R$ satisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. Although it is not hard to verify that every integral domain satisfying ACCP is atomic, examples of atomic domains that do not satisfy ACCP are notoriously hard to construct. The first of such examples was constructed by A. Grams back in 1974. In this paper we delve into the class of atomic domains that do not satisfy ACCP. To better understand this class, we introduce the notion of weak-ACCP domains, which generalizes that of integral domains satisfying ACCP. Strongly atomic domains were introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. It turns out that every weak-ACCP domain is strongly atomic, and so we introduce a taxonomic classification on our class of interest: ACCP implies weak-ACCP, which implies strong atomicity, which implies atomicity. We study this chain of implications, putting special emphasis on the weak-ACCP property. This allows us to provide new examples of atomic domains that do not satisfy ACCP.
Forward citations
Cited by 4 Pith papers
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On finitary power monoids of linearly orderable monoids
For linearly orderable monoids, quasi-ACCP and almost ACCP ascend to finitary power monoids, while atomicity, near atomicity, and quasi-atomicity do not; atomic power monoids are characterized as those from atomic MCD...
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Arithmetic properties encoded in undermonoids
A monoid is hereditarily atomic exactly when all its undermonoids are atomic, this forces the ACCP, and torsion abelian groups are precisely the monoids all of whose undermonoids are half- or length-factorial.
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On the atomicity of one-dimensional monoid algebras
For every rational q in (0,1), except reciprocals of integers, whose denominator is odd, the monoid algebra Q[M_q] is atomic.
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On the additive structure of algebraic valuations of polynomial semirings II
For algebraic α, M_α is antimatter iff isomorphic to a finite product of valuation monoids, which for α in (0,1) hold precisely when α^{-1} is a Perron number with no other positive conjugates; such α are dense in (0,1).
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