On noncommutative bounded factorization domains and prime rings
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866917195184865280 |
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| author | Bell, Jason P. Brown, Ken Nazemian, Zahra Smertnig, Daniel |
| author_facet | Bell, Jason P. Brown, Ken Nazemian, Zahra Smertnig, Daniel |
| contents | A ring has bounded factorizations if every cancellative nonunit $a \in R$ can be written as a product of atoms and there is a bound $λ(a)$ on the lengths of such factorizations. The bounded factorization property is one of the most basic finiteness properties in the study of non-unique factorizations. Every commutative noetherian domain has bounded factorizations, but it is open whether such a result holds in the noncommutative setting. We provide sufficient conditions for a noncommutative noetherian prime ring to have bounded factorizations. Moreover, we construct a (noncommutative) finitely presented semigroup algebra that is an atomic domain but does not satisfy the ascending chain condition on principal right or left ideals (ACCP), whence it does not have bounded factorizations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_10115 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On noncommutative bounded factorization domains and prime rings Bell, Jason P. Brown, Ken Nazemian, Zahra Smertnig, Daniel Rings and Algebras Primary 16P40, Secondary 13F15, 16E65, 20M13 A ring has bounded factorizations if every cancellative nonunit $a \in R$ can be written as a product of atoms and there is a bound $λ(a)$ on the lengths of such factorizations. The bounded factorization property is one of the most basic finiteness properties in the study of non-unique factorizations. Every commutative noetherian domain has bounded factorizations, but it is open whether such a result holds in the noncommutative setting. We provide sufficient conditions for a noncommutative noetherian prime ring to have bounded factorizations. Moreover, we construct a (noncommutative) finitely presented semigroup algebra that is an atomic domain but does not satisfy the ascending chain condition on principal right or left ideals (ACCP), whence it does not have bounded factorizations. |
| title | On noncommutative bounded factorization domains and prime rings |
| topic | Rings and Algebras Primary 16P40, Secondary 13F15, 16E65, 20M13 |
| url | https://arxiv.org/abs/2206.10115 |