Ore extensions of abelian groups with operators
Uloženo v:
| Název: | Ore extensions of abelian groups with operators |
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| Autoři: | Bäck, Per, 1987, Lundström, Patrik, Öinert, Johan, Richter, Johan |
| Zdroj: | Journal of Algebra. 686:176-194 |
| Témata: | Ore group extension, Ore module extension, Noetherian group, Noetherian module, Vandermonde’s identity, Leibniz’s identity, Hilbert’s basis theorem, Mathematics/Applied Mathematics, matematik/tillämpad matematik |
| Popis: | Given a set A and an abelian group B with operators in A, we introduce the Ore group extension B[x; σ_B, δ_B] as the additive group B[x], with A[x] as a set of operators. Here, the action of A[x] on B[x] is defined by mimicking the multiplication used in the classical case where A and B are the same ring. We derive generalizations of Vandermonde's and Leibniz's identities for this construction, and they are then used to establish associativity criteria. Additionally, we prove a version of Hilbert's basis theorem for this structure, under the assumption that the action of A on B is what we call weakly s-unital. Finally, we apply these results to the case where B is a left module over a ring A, and specifically to the case where A and B coincide with a non-associative ring which is left distributive but not necessarily right distributive. |
| Popis souboru: | |
| Přístupová URL adresa: | https://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-68798 https://doi.org/10.1016/j.jalgebra.2025.06.042 |
| Databáze: | SwePub |
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Nájsť tento článok vo Web of Science | Abstrakt: | Given a set A and an abelian group B with operators in A, we introduce the Ore group extension B[x; σ_B, δ_B] as the additive group B[x], with A[x] as a set of operators. Here, the action of A[x] on B[x] is defined by mimicking the multiplication used in the classical case where A and B are the same ring. We derive generalizations of Vandermonde's and Leibniz's identities for this construction, and they are then used to establish associativity criteria. Additionally, we prove a version of Hilbert's basis theorem for this structure, under the assumption that the action of A on B is what we call weakly s-unital. Finally, we apply these results to the case where B is a left module over a ring A, and specifically to the case where A and B coincide with a non-associative ring which is left distributive but not necessarily right distributive. |
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| ISSN: | 00218693 1090266X |
| DOI: | 10.1016/j.jalgebra.2025.06.042 |