Goldbach theorems for group semidomains

Abstract

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 ${\mathcal{A}}_{+}(S)=S^{\times }$ holds.

14 pages