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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.00590 |
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| _version_ | 1866909437425352704 |
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| author | Li, Eddy Mopuri, Advaith Zhang, Charles |
| author_facet | Li, Eddy Mopuri, Advaith Zhang, Charles |
| contents | A semidomain is a subsemiring of an integral domain. We call a semidomain $S$ additively reduced if $0$ is the only invertible element of the monoid $(S, +)$, while we say that $S$ is additively Furstenberg if every non-invertible element of $(S,+)$ can be expressed as the sum of an atom and an element of $S$. In this paper, we study a variant of the Goldbach conjecture within the framework of group semidomains $S[G]$ and group series semidomains $S[\![G]\!]$, where $S$ is both an additively reduced and additively Furstenberg semidomain and $G$ is a torsion-free abelian group. In particular, we show that every non-constant polynomial expression in $S[G]$ can be written as the sum of at most two irreducibles if and only if the condition $\mathscr{A}_+(S) = S^\times$ holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00590 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Goldbach theorems for group semidomains Li, Eddy Mopuri, Advaith Zhang, Charles Commutative Algebra Primary: 11P32, 16Y60, Secondary: 20M13 A semidomain is a subsemiring of an integral domain. We call a semidomain $S$ additively reduced if $0$ is the only invertible element of the monoid $(S, +)$, while we say that $S$ is additively Furstenberg if every non-invertible element of $(S,+)$ can be expressed as the sum of an atom and an element of $S$. In this paper, we study a variant of the Goldbach conjecture within the framework of group semidomains $S[G]$ and group series semidomains $S[\![G]\!]$, where $S$ is both an additively reduced and additively Furstenberg semidomain and $G$ is a torsion-free abelian group. In particular, we show that every non-constant polynomial expression in $S[G]$ can be written as the sum of at most two irreducibles if and only if the condition $\mathscr{A}_+(S) = S^\times$ holds. |
| title | Goldbach theorems for group semidomains |
| topic | Commutative Algebra Primary: 11P32, 16Y60, Secondary: 20M13 |
| url | https://arxiv.org/abs/2412.00590 |