Hostname: page-component-6bf8c574d5-t27h7 Total loading time: 0 Render date: 2025-02-17T00:46:19.920Z Has data issue: false hasContentIssue false
  • English
  • Français

A characterisation of atomicity

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

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bell, J. P., Brown, K., Nazemian, Z. and Smertnig, D.. On noncommutative bounded factorisation domains and prime rings. J. Algebra 622 (May 2023), 404–449.CrossRefGoogle Scholar
Boynton, J. G. and Coykendall, J.. An example of an atomic pullback without the ACCP. J. Pure Appl. Algebra 223 (2019), 619–625.CrossRefGoogle Scholar
Chapman, S. T.. A tale of two monoids: a friendly introduction to nonunique factorisations. Math. Mag. 87 (2014), No. 3, 163–173.Google Scholar
Chapman, S. T., Gotti, F. and Gotti, M.. Factorisation invariants of Puiseux monoids generated by geometric sequences. Comm. Algebra 48 (2020), No. 1, 380–396.Google Scholar
Chapman, S. T., Gotti, F. and Gotti, M.. When is a Puiseux monoid atomic? Amer. Math. Monthly 128 (2021), No. 4, 302–321.Google Scholar
Cohn, P. M.. Free ideal rings. J. Algebra 1 (1964), 47–69.CrossRefGoogle Scholar
Cohn, P. M.. Torsion modules over free ideal rings. Proc. London Math. Soc. III. Ser. 17 (1967), 577–599.CrossRefGoogle Scholar
Cohn, P. M.. Bezout rings and their subrings, Math. Proc. Camb. Phil. Soc. 64 (1968), No. 2, 251–264.Google Scholar
Cohn, P. M.. Unique factorisation domains. Amer. Math. Monthly 80 (1973), No. 1, 1–18.Google Scholar
Cohn, P. M.. Free Ideal Rings and Localisation in General Rings. New Math. Monogr. 3 (Cambridge University Press, 2006).CrossRefGoogle Scholar
Cossu, L. and Tringali, S.. Abstract factorization theorems with applications to idempotent factorisations. Israel J. Math., to appear (https://arxiv.org/abs/2108.12379).Google Scholar
Cossu, L. and Tringali, S.. Factorisation under local finiteness conditions. J. Algebra, to appear (https://arxiv.org/abs/2208.05869).Google Scholar
Coykendall, J. and Gotti, F.. On the atomicity of monoid algebras. J. Algebra 539 (2019), 138–151.CrossRefGoogle Scholar
Davey, B. A. and Priestley, H. A.. Introduction to Lattices and Order (2nd edition). (Cambridge University Press, 2002).CrossRefGoogle Scholar
Fan, Y. and Tringali, S.. Power monoids: a bridge between factorisation theory and arithmetic combinatorics. J. Algebra 512 (Oct. 2018), 252–294.Google Scholar
Geroldinger, A. and Halter–Koch, F.. Non-Unique Factorisations. Algebraic, Combinatorial and Analytic Theory. Pure Appl. Math. 278 (Chapman and Hall/CRC: Boca Raton, FL, 2006).CrossRefGoogle Scholar
Geroldinger, A. and Zhong, Q.. Factorisation theory in commutative monoids. Semigroup Forum 100 (2020), 22–51.CrossRefGoogle Scholar
Gilmer, R.. Commutative Semigroup Rings. Chicago Lect. Math. XII, 380 pp. (University of Chicago Press: Chicago, IL, 1984).Google Scholar
Gotti, F. and Li, B.. Atomic semigroup rings and the ascending chain condition on principal ideals. Proc. Amer. Math. Soc. 151 (2023), 2291–2302.Google Scholar
Grams, A.. Atomic rings and the ascending chain condition for principal ideals. Math. Proc. Camb. Phil. Soc. 75 (1974), No. 3, 321–329.Google Scholar
Howie, J. M.. Fundamentals of Semigroup Theory. London Math. Soc. Monogr. Ser. 12 (Oxford University Press, 1995).Google Scholar
Roitman, M.. Polynomial extensions of atomic domains. J. Pure Appl. Algebra 87 (1993), No. 2, 187–199.Google Scholar
Tringali, S.. An abstract factorisation theorem and some applications. J. Algebra 602 (July 2022), 352–380.CrossRefGoogle Scholar
A. Zaks. Atomic rings without a.c.c. on principal ideals. J. Algebra 80 (1982), 223–231.Google Scholar