Computing the unit group of a commutative finite $\mathbb{Z}$-algebra
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909650746605568 |
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| author | Kreuzer, Martin Walsh, Florian |
| author_facet | Kreuzer, Martin Walsh, Florian |
| contents | For a commutative finite $\mathbb{Z}$-algebra, i.e., for a commutative ring $R$ whose additive group is finitely generated, it is known that the group of units of $R$ is finitely generated, as well. Our main results are algorithms to compute generators and the structure of this group. This is achieved by reducing the task first to the case of reduced rings, then to torsion-free reduced rings, and finally to an order in a reduced ring. The simplified cases are treated via a calculation of exponent lattices and various algorithms to compute the minimal primes, primitive idempotents, and other basic objects. All algorithms have been implemented and are available as a SageMath package. Whenever possible, the time complexity of the described methods is tracked carefully. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07002 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing the unit group of a commutative finite $\mathbb{Z}$-algebra Kreuzer, Martin Walsh, Florian Commutative Algebra Group Theory Rings and Algebras 68W30 (Primary) 20F05 13P99 16Z05 (Secondary) For a commutative finite $\mathbb{Z}$-algebra, i.e., for a commutative ring $R$ whose additive group is finitely generated, it is known that the group of units of $R$ is finitely generated, as well. Our main results are algorithms to compute generators and the structure of this group. This is achieved by reducing the task first to the case of reduced rings, then to torsion-free reduced rings, and finally to an order in a reduced ring. The simplified cases are treated via a calculation of exponent lattices and various algorithms to compute the minimal primes, primitive idempotents, and other basic objects. All algorithms have been implemented and are available as a SageMath package. Whenever possible, the time complexity of the described methods is tracked carefully. |
| title | Computing the unit group of a commutative finite $\mathbb{Z}$-algebra |
| topic | Commutative Algebra Group Theory Rings and Algebras 68W30 (Primary) 20F05 13P99 16Z05 (Secondary) |
| url | https://arxiv.org/abs/2405.07002 |