发表机构
Mindanao State University-Iligan Institute of Technology; Blekinge Institute of Technology（棉兰老州立大学伊利甘理工学院; 布莱金厄理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为非结合斜幂级数环和斜洛朗级数环建立希尔伯特基定理，通过引入拟结合性质给出Noetherian条件，并指出一般非结合情形下定理不成立。
AI 中文摘要
本文中，我们给出单位非结合环$R$和加性映射$\sigma: R\longrightarrow R$的条件，使得非结合斜幂级数环（记为$R[[X;\sigma]]$）和非结合斜洛朗级数环（记为$R((X;\sigma))$）分别为左（或右）Noetherian环。因此，我们分别建立了$R[[X;\sigma]]$和$R((X;\sigma))$的左（或右）Noetherian性的希尔伯特基定理版本。我们还给出了一个反例，表明在非结合情形下希尔伯特基定理一般不成立。在建立$R[[X;\sigma]]$和$R((X;\sigma))$的希尔伯特基定理时，我们引入并利用了$R$的右（或左）拟结合性质，该性质等价于对所有$r\in R$（或$c\in R$），$rR$（或$Rc$）是单侧理想。
英文摘要
In this paper, we give conditions on a unital non-associative ring $R$ and additive map $σ: R\longrightarrow R$ so that the non-associative skew power series ring, denoted by $R[[X;σ]]$ and the non-associative skew Laurent series ring denoted by $R((X;σ))$ are left (resp. right) Noetherian, respectively. Consequently, we establish versions of Hilbert Basis Theorem of $R[[X;σ]]$ and $R((X;σ))$, respectively, for left (resp. right) Noetherianity . We also give a counterexample showing that the Hilbert Basis Theorem does not hold in general in the non-associative case. In establishing the Hilbert Basis Theorems for $R[[X;σ]]$ and $R((X;σ))$, we introduce and utilize the right (resp. left) quasi-associative property of $R$, which is equivalent to $rR$ (resp. $Rc$) being one-sided ideals for all $r\in R$ (resp. $c\in R$).
Comments36 pages
