A characterization of atomicity
Abstract
In [Math. Proc. Cambridge Philos. Soc. 64 (1968), 251-264], 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 provided by A. Grams in [Math. Proc. Cambridge Philos. Soc. 75 (1974), 321-329]: Every commutative domain with the ACCP is atomic, but not vice versa. This has led to the question of finding a sensible (ideal-theoretic) characterization of atomicity. The question (explicitly stated on p. 3 of A. Geroldinger and F. Halter-Koch's 2006 monograph on factorization) is still open. We settle it using the language of monoids and preorders.
Cite
@article{arxiv.2209.05238,
title = {A characterization of atomicity},
author = {Salvatore Tringali},
journal= {arXiv preprint arXiv:2209.05238},
year = {2023}
}
Comments
7 pp., no figures. Shortened the title, fixed a detail in Remark 2.2(2), and added Examples 2.7. Final version to appear in Math. Proc. Cambridge Philos. Soc