On the atomicity of one-dimensional monoid algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909403667496960 |
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| author | Panpaliya, Ishan |
| author_facet | Panpaliya, Ishan |
| contents | 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 $S := ((0, 1) \cap \mathbb{Q}) \setminus {\mathbb{N}}^{-1}_{> 1}$, atomicity ascends from the exponentially cyclic Puiseux monoid $M_q$ 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 $\mathbb{Q} \left[M_{3/4} \right]$ is atomic, marking the first progress towards settling this conjecture. We strengthen this result and prove that $\mathbb{Q}[M_q]$ is atomic for all $q \in S$ having an odd denominator. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_16151 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the atomicity of one-dimensional monoid algebras Panpaliya, Ishan Commutative Algebra Primary: 13F15, 13A05, Secondary: 20M25, 13B22 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 $S := ((0, 1) \cap \mathbb{Q}) \setminus {\mathbb{N}}^{-1}_{> 1}$, atomicity ascends from the exponentially cyclic Puiseux monoid $M_q$ 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 $\mathbb{Q} \left[M_{3/4} \right]$ is atomic, marking the first progress towards settling this conjecture. We strengthen this result and prove that $\mathbb{Q}[M_q]$ is atomic for all $q \in S$ having an odd denominator. |
| title | On the atomicity of one-dimensional monoid algebras |
| topic | Commutative Algebra Primary: 13F15, 13A05, Secondary: 20M25, 13B22 |
| url | https://arxiv.org/abs/2411.16151 |