关于斜多项式环中子扩展的均匀维度
On the uniform dimension of subextensions in skew polynomial rings
AI总结：
本文研究了斜多项式环中子扩展的均匀维度性质，揭示了在不同变量类型下子扩展行为的差异，并给出了相关结论。
摘要：
本文研究了在斜多项式环 \mbox{$ \mathbb{S}=R[ \mathbf{\mathrm{X}}; \mathbfα , \mathbfδ ]$} 中子扩展的非必要有限均匀维度及相关概念的不变性，该环为双射类型，变量集为良序集。当系数环具有足够的均匀左理想时，在交换变量情况下，我们证明了关于多项式环的经典结果可以扩展到由任何标准项生成的斜Laurent多项式环 \mbox{$ \mathbb{S}=R[ \mathbf{\mathrm{X}}^{\pm1}; \mathbfα]$} 的子扩展。在非交换变量情况下，情况更为复杂；容易形成的多项式样子环可以表现出非常奇特的行为。我们提供了易于构造的（半）素左Goldie斜多项式环的例子，该环为双射类型，包含一个同构于自由非交换多项式环的单调子环。然后我们研究了所谓的子类 \emph{本质上特殊的子扩展}，并证明了它们的均匀维度及相关概念的保持。
英文摘要：
This work investigates the invariance of the non-necessarily finite uniform dimension and related concepts for subextensions in skew polynomial rings \mbox{$ \mathbb{S}=R[ \mathbf{\mathrm{X}}; \mathbfα , \mathbfδ ]$} of bijective type over a well-ordered set of variables. When the coefficient ring has enough uniform left ideals, in the commuting variables case we show that classical results on this topic for polynomial rings extend to subextensions of skew Laurent polynomial rings \mbox{$ \mathbb{S}=R[ \mathbf{\mathrm{X}}^{\pm1}; \mathbfα]$}, generated over $R$ by any family of (standard) terms. The situation in the non-commuting variables context is more complex; easily formed polynomial-like subrings can behave very oddly from the ambient ring. We provide easy examples of a (semi)prime left Goldie skew polynomial ring of bijective type containing a monoid subring isomorphic to a free non-commutative polynomial ring. We then study the so-called subclass of \emph{essentially special subextensions} and obtain for them the preservation of the uniform dimension and related concepts.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
