2024/11/25 by Ishan Panpaliya, Panpaliya, Ishan
Mathematics · #13A05 #13B22 #Commutative Algebra (math.AC) #FOS: Mathematics #Primary: 13F15 #Rings, Modules, and Algebras #Secondary: 20M25
paper · pdf · doi:10.48550/arxiv.2411.16151
openalex publication_date 2024/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The ascending chain condition on principal ideals (ACCP) is almost always complementary to atomicity within integral domains: in fact, Cohn initially stated that these two conditions were equivalent. This assertion has been shown to be false, however most counterexamples require technical algebraic constructions. In 2017, Gotti conjectured that for every q in the set S := ((0, 1) ∩ ℚ) ∖ ℕ-1> 1, atomicity ascends from the exponentially cyclic Puiseux monoid Mq to its monoid algebra over the field of rationals. If this conjecture were true, it would provide an extremely wide class of atomic domains of Krull dimension one not satisfying the ACCP, and so would be perhaps the simplest possible such examples. Bu et al. recently proved that the monoid algebra ℚ [M3/4 ] is atomic, marking the first progress towards settling this conjecture. We strengthen this result and prove that ℚ[Mq] is atomic for all q ∈ S having an odd denominator.