标题： 关于群半域的哥德巴赫定理 显示英文标题

标题： Goldbach theorems for group semidomains

摘要： 半域是一个整环的子半环。 我们称半域$S$在加法上是约化的，如果$0$是独异点$(S, +)$的唯一可逆元，而我们说$S$在加法上是弗斯特纳格的，如果$(S,+)$的每个不可逆元都可以表示为一个素元和一个元素的和，该元素属于$S$。 在本文中，我们在群半环$S[G]$和群级数半环$S[\![G]\!]$的框架内研究了哥德巴赫猜想的一个变体，其中$S$既是可加约化的也是可加的 Furstenberg 半环，而$G$是一个无挠的阿贝尔群。特别地，我们证明了在$S[G]$中的每一个非常值多项式表达式都可以表示为最多两个不可约元的和，当且仅当条件$\mathscr{A}_+(S) = S^\times$成立。 显示英文摘要

摘要： 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.