发表机构
Roma Tre University（罗马第三大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对一类非加法主序数$\alpha$，利用赋值理论的$RV$工具，建立了不被单项式整除且支撑集序型为$\omega^{\alpha}$或$\omega^{\alpha}+1$的广义幂级数环元素的唯一分解性。
AI 中文摘要
形如$K((G^{\le 0}))$的环中的分解因单项式的可除性表现出病态行为。仍待解决的问题是：这是否是唯一的阻碍，以及由所有单项式生成的理想的商是否为唯一分解整环。在本研究之前，除不可约元素外的唯一分解实例直接源于不被任何单项式整除且支撑集序型为$\omega$或$\omega+1$的元素的素性。本文使用赋值理论中的$RV$工具作为关键要素，对不被任何单项式整除且支撑集序型为$\omega^{\alpha}$或$\omega^{\alpha}+1$的$K((\mathbb{R}^{\le 0}))$中的元素，在一大类非加法主序数$\alpha$下建立了唯一分解性。
英文摘要
Factorization in rings of the form $K((G^{\le 0} )) $ exhibits pathological behavior due to divisibility by monomials. It remains open whether this is the only obstruction and whether the quotient by the ideal generated by all monomials is a unique factorization domain. Prior to this work, the only instances of unique factorization beyond the irreducible elements followed directly from the primality of elements which aren't divisible by any monomial and whose support has order type $ω$ or $ω+1$. Using the $RV$ tool from valuation theory as a key ingredient, we establish unique factorization for elements of $ K((\mathbb{R}^{\le 0}))$ which aren't divisible by any monomial and whose support has order type $ω^α $ or $ω^α+1 $ for a large class of non additively principal ordinals $α$.
