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A characterisation of atomicity

Part of: General commutative ring theory Semigroups Chain conditions, finiteness conditions Chain conditions, growth conditions, and other forms of finiteness

Published online by Cambridge University Press:  24 April 2023

SALVATORE TRINGALI
SALVATORE TRINGALI*
Affiliation:
School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei province, 050024 China. e-mail: salvo.tringali@gmail.com
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Abstract

In a 1968 issue of the Proceedings, P. M. Cohn famously claimed that a commutative domain is atomic if and only if it satisfies the ascending chain condition on principal ideals (ACCP). Some years later, a counterexample was however provided by A. Grams, who showed that every commutative domain with the ACCP is atomic, but not vice versa. This has led to the problem of finding a sensible (ideal-theoretic) characterisation of atomicity.

The question (explicitly stated on p. 3 of A. Geroldinger and F. Halter–Koch’s 2006 monograph on factorisation) is still open. We settle it here by using the language of monoids and preorders.

MSC classification

Primary: 13A05: Divisibility; factorizations 13E99: None of the above, but in this section 16P70: Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence 20M10: General structure theory 20M13: Arithmetic theory of monoids
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 175 , Issue 2 , September 2023 , pp. 459 - 465
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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