The domain and prime properties for Koszul rings and algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916329228861440 |
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| author | Reyes, Manuel L. Rogalski, Daniel |
| author_facet | Reyes, Manuel L. Rogalski, Daniel |
| contents | We establish a technique to prove that a Koszul graded ring is prime or a domain using information about its Koszul dual. This is based on a general categorical result that expands on methods of J.Y. Guo, which proves that certain orbital rings are prime or domains. We apply this method to prove that if $A = kQ/I$ is a Koszul twisted Calabi-Yau algebra of dimension 2, such that $Q$ is connected with every vertex having outdegree at least 2, then $A$ is a prime piecewise domain. In particular, the preprojective algebra of a connected quiver whose underlying graph has minimum degree at least 2 is a prime piecewise domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13119 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The domain and prime properties for Koszul rings and algebras Reyes, Manuel L. Rogalski, Daniel Rings and Algebras Category Theory Representation Theory Primary: 16E65, 16N60, 16S37, 16U10, Secondary: 16E30, 16S99 We establish a technique to prove that a Koszul graded ring is prime or a domain using information about its Koszul dual. This is based on a general categorical result that expands on methods of J.Y. Guo, which proves that certain orbital rings are prime or domains. We apply this method to prove that if $A = kQ/I$ is a Koszul twisted Calabi-Yau algebra of dimension 2, such that $Q$ is connected with every vertex having outdegree at least 2, then $A$ is a prime piecewise domain. In particular, the preprojective algebra of a connected quiver whose underlying graph has minimum degree at least 2 is a prime piecewise domain. |
| title | The domain and prime properties for Koszul rings and algebras |
| topic | Rings and Algebras Category Theory Representation Theory Primary: 16E65, 16N60, 16S37, 16U10, Secondary: 16E30, 16S99 |
| url | https://arxiv.org/abs/2407.13119 |