Primes and absolutely or non-absolutely irreducible elements in atomic domains
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909986938945536 |
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| author | Fadinger, Victor Frisch, Sophie Nakato, Sarah Smertnig, Daniel Windisch, Daniel |
| author_facet | Fadinger, Victor Frisch, Sophie Nakato, Sarah Smertnig, Daniel Windisch, Daniel |
| contents | We give examples of atomic integral domains satisfying each of the eight logically possible combinations of existence or non-existence of the following kinds of elements: 1) primes, 2) absolutely irreducible elements that are not prime, and 3) irreducible elements that are not absolutely irreducible. A non-zero non-unit is called absolutely irreducible (or, a strong atom) if every one of its powers factors uniquely into irreducibles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01051 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Primes and absolutely or non-absolutely irreducible elements in atomic domains Fadinger, Victor Frisch, Sophie Nakato, Sarah Smertnig, Daniel Windisch, Daniel Commutative Algebra 13A05, 11S05, 11R09, 13B25, 13F20, 11C08 We give examples of atomic integral domains satisfying each of the eight logically possible combinations of existence or non-existence of the following kinds of elements: 1) primes, 2) absolutely irreducible elements that are not prime, and 3) irreducible elements that are not absolutely irreducible. A non-zero non-unit is called absolutely irreducible (or, a strong atom) if every one of its powers factors uniquely into irreducibles. |
| title | Primes and absolutely or non-absolutely irreducible elements in atomic domains |
| topic | Commutative Algebra 13A05, 11S05, 11R09, 13B25, 13F20, 11C08 |
| url | https://arxiv.org/abs/2411.01051 |