Divide and Transfer: Non-Unique Factorizations Beyond Commutativity
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866918325077934080 |
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| author | Smertnig, Daniel |
| author_facet | Smertnig, Daniel |
| contents | Unique factorization fails in many rings and monoids, but divisor and transfer homomorphisms provide tools to understand non-unique factorizations. In this expository article, we first explore these notions in the classical setting of commutative Dedekind domains, where monoids of zero-sum sequences appear as a natural combinatorial model. We then adapt these ideas to the setting of noncommutative Dedekind prime rings using module-theoretic methods. Going a step further, we discuss Rump and Yang's recent divisor theory for ideals in hereditary noetherian prime rings, where divisors can be visualized in a diagrammatic calculus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06222 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Divide and Transfer: Non-Unique Factorizations Beyond Commutativity Smertnig, Daniel Rings and Algebras Unique factorization fails in many rings and monoids, but divisor and transfer homomorphisms provide tools to understand non-unique factorizations. In this expository article, we first explore these notions in the classical setting of commutative Dedekind domains, where monoids of zero-sum sequences appear as a natural combinatorial model. We then adapt these ideas to the setting of noncommutative Dedekind prime rings using module-theoretic methods. Going a step further, we discuss Rump and Yang's recent divisor theory for ideals in hereditary noetherian prime rings, where divisors can be visualized in a diagrammatic calculus. |
| title | Divide and Transfer: Non-Unique Factorizations Beyond Commutativity |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2602.06222 |