Atomic semigroup rings and the ascending chain condition on principal ideals

Gotti, Felix; Li, Bangzheng

doi:10.1090/proc/16295

Atomic semigroup rings and the ascending chain condition on principal ideals
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by Felix Gotti and Bangzheng Li;

Proc. Amer. Math. Soc. 151 (2023), 2291-2302

DOI: https://doi.org/10.1090/proc/16295

Published electronically: March 21, 2023

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Abstract:

An integral domain is called atomic if every nonzero nonunit element factors into irreducibles. On the other hand, an integral domain is said to satisfy the ascending chain condition on principal ideals (ACCP) if every ascending chain of principal ideals stabilizes. It was asserted by P. Cohn back in the sixties that every atomic domain satisfies the ACCP, but such an assertion was refuted by A. Grams in the seventies with a neat counterexample. Still, atomic domains without the ACCP are notoriously elusive, and just a few classes have been found since Grams’ first construction. In the first part of this paper, we generalize Grams’ construction to provide new classes of atomic domains without the ACCP. In the second part of this paper, we construct a new class of atomic semigroup rings without the ACCP.

References

Jason G. Boynton and Jim Coykendall, An example of an atomic pullback without the ACCP, J. Pure Appl. Algebra 223 (2019), no. 2, 619–625. MR 3850560, DOI 10.1016/j.jpaa.2018.04.010
Scott T. Chapman, Felix Gotti, and Marly Gotti, Factorization invariants of Puiseux monoids generated by geometric sequences, Comm. Algebra 48 (2020), no. 1, 380–396. MR 4060036, DOI 10.1080/00927872.2019.1646269
Scott T. Chapman, Felix Gotti, and Marly Gotti, When is a Puiseux monoid atomic?, Amer. Math. Monthly 128 (2021), no. 4, 302–321. MR 4234728, DOI 10.1080/00029890.2021.1865064
P. M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc. 64 (1968), 251–264. MR 222065, DOI 10.1017/s0305004100042791
Jim Coykendall and Felix Gotti, On the atomicity of monoid algebras, J. Algebra 539 (2019), 138–151. MR 3995239, DOI 10.1016/j.jalgebra.2019.07.033
Alfred Geroldinger, Felix Gotti, and Salvatore Tringali, On strongly primary monoids, with a focus on Puiseux monoids, J. Algebra 567 (2021), 310–345. MR 4159257, DOI 10.1016/j.jalgebra.2020.09.019
Alfred Geroldinger and Franz Halter-Koch, Non-unique factorizations, Pure and Applied Mathematics (Boca Raton), vol. 278, Chapman & Hall/CRC, Boca Raton, FL, 2006. Algebraic, combinatorial and analytic theory. MR 2194494, DOI 10.1201/9781420003208
Robert Gilmer, Commutative semigroup rings, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1984. MR 741678
Felix Gotti, On semigroup algebras with rational exponents, Comm. Algebra 50 (2022), no. 1, 3–18. MR 4370409, DOI 10.1080/00927872.2021.1949018
Felix Gotti and Bangzheng Li, Divisibility in rings of integer-valued polynomials, New York J. Math. 28 (2022), 117–139. MR 4367411
Anne Grams, Atomic rings and the ascending chain condition for principal ideals, Proc. Cambridge Philos. Soc. 75 (1974), 321–329. MR 340249, DOI 10.1017/s0305004100048532
H. Kim: Factorization in monoid domains. PhD Dissertation, The University of Tennessee, Knoxville, 1998.
Hwankoo Kim, Factorization in monoid domains, Comm. Algebra 29 (2001), no. 5, 1853–1869. MR 1837946, DOI 10.1081/AGB-100002153
Moshe Roitman, Polynomial extensions of atomic domains, J. Pure Appl. Algebra 87 (1993), no. 2, 187–199. MR 1224218, DOI 10.1016/0022-4049(93)90122-A
Abraham Zaks, Atomic rings without a.c.c. on principal ideals, J. Algebra 74 (1982), no. 1, 223–231. MR 644228, DOI 10.1016/0021-8693(82)90015-1