Goldbach theorems for group semidomains

TL;DR

This paper explores a variant of the Goldbach conjecture within the algebraic framework of group semidomains, establishing conditions under which polynomial expressions can be decomposed into sums of irreducibles.

Contribution

It introduces the study of Goldbach-type theorems for group semidomains and characterizes when polynomial expressions decompose into sums of irreducibles.

Findings

Every non-constant polynomial in $S[G]$ can be expressed as the sum of at most two irreducibles if and only if $\\mathscr{A}_+(S) = S^\times$.

The paper extends Goldbach conjecture concepts to algebraic structures called group semidomains.

Conditions for polynomial decompositions are characterized in terms of properties of the semidomain $S$.

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