F-诺特环与斜量子环扩张
F-Noetherian Rings and Skew Quantum Ring Extensions
摘要：
一个环R称为F-诺特环，如果R的每个有限子集都包含在R的一个（左和右）诺特子环中。例如，每一个交换环是紧F-诺特环，因为R的每一个有限子集生成R的一个诺特子环。F-诺特环有许多有趣的线性代数性质，我们称之为完全强秩条件，完全稳定有限，更一般的基本条件。我们还研究了F-诺特环的一些基本的环论性质，如F-诺特环的局部化. F-Noether性质在某些\emph{skew}量子环扩张下保持不变，这些扩张包括一些迭代Ore扩张、一些斜Laurent扩张和一些量子几乎正规化扩张.例如，令R= S[ x_1，...，x_n ]是{在一个子环S上}生成的环\textit，使得（1）对于i < j，对于某些单元q_{ji} \in S，\[ x_j x_i -q_{ji} x_i x_j \in S [ x_1, ..., x_{j-1}] + Sx_j \]，（2）对于所有i，Sx_i +S= S+ x_i S，以及（3）每个x_i与S的一个子环A交换，使得S作为A上的环生成。如果S是F-诺特的，那么R也是F-诺特的。我们还讨论了与量子群\mathcal{O}_q（G）有关的斜二次扩张，这里G是一个连通的复半单代数群.最后，我们给出了F-Noether环k上的量子群如\mathcal{O}_q（M_n（k））的许多例子和一些推广，其中每个变量x_{ij} \textit{may not commute}的元素为k。
英文摘要：
A ring R shall be called F-noetherian if every finite subset of R is contained in a (left and right) noetherian subring of R . For example, every commutative ring is tightly F-noetherian in the sense that every finite subset of R generates a noetherian subring of R . F-noetherian rings have many interesting linear algebra properties which we refer to as the full strong rank condition, fully stably finite, and more generally the basic condition. We also study some basic ring-theoretic properties of F-noetherian rings such as localizations of F-noetherian rings. The F-noetherian property is preserved under some \emph{skew} quantum ring extensions including some iterated Ore extensions, some skew-Laurent extensions, and some quantum almost-normalizing extensions. For example, let R= S[ x_1, ..., x_n ] be a finitely generated ring \textit {over a subring S} such that (1) for i < j, \[ x_j x_i -q_{ji} x_i x_j \in S [ x_1, ..., x_{j-1}] + Sx_j \] for some units q_{ji} \in S, (2) for all i, Sx_i +S= S+ x_i S, and (3) each x_i commutes with a subring A of S such that S is finitely generated as a ring over A . Then, if S is F-noetherian, so is R . We also discuss some skew quadratic extensions related to the quantum group \mathcal{O}_q(G) where G is a connected complex semisimple algebraic group. Finally, we show many examples and some generalizations of some quantum groups like \mathcal{O}_q(M_n(k)) over an F-noetherian ring k where each variable x_{ij} \textit{may not commute} with the elements of k .
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。
