Ore Extensions of Abelian Groups with Operators
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909754035535872 |
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| author | Bäck, Per Lundström, Patrik Öinert, Johan Richter, Johan |
| author_facet | Bäck, Per Lundström, Patrik Öinert, Johan Richter, Johan |
| contents | Given a set $A$ and an abelian group $B$ with operators in $A$, in the sense of Krull and Noether, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_16761 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ore Extensions of Abelian Groups with Operators Bäck, Per Lundström, Patrik Öinert, Johan Richter, Johan Rings and Algebras Representation Theory 16S36, 16W22, 16W70, 17A99, 17D99, 20K27 Given a set $A$ and an abelian group $B$ with operators in $A$, in the sense of Krull and Noether, 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. |
| title | Ore Extensions of Abelian Groups with Operators |
| topic | Rings and Algebras Representation Theory 16S36, 16W22, 16W70, 17A99, 17D99, 20K27 |
| url | https://arxiv.org/abs/2410.16761 |