Fox代数、局部化与自由多项式的因式分解
The Fox algebra, Localization and factorizations of free polynomials
AI总结：
本文建立 Cohn、Leavitt 与 Jacobson 局部化的联系，并利用自由群的 Fox 代数发展非零增广自由多项式的因式分解理论，给出因式分解与合成链的双射对应。
摘要：
我们讨论了以下三者之间的相互关系：满秩方阵的 Cohn 局部化；一行的 Leavitt 局部化；以及拟正则元素的 Jacobson 拟逆。后一种 Jacobson 局部化会在关于某种理想拓扑为 Hausdorff 拓扑空间的环中自然且容易地出现，这也指出了其与特定 Gabriel 局部化之间的联系。作为主要结果和应用，我们发展了一套关于域上具有非零增广的自由多项式的因式分解理论。其灵感来自与 Mantese 的合作工作中针对常数项位于非交换变量中的多项式所给出的因式分解理论。本研究的基本工具是利用一行自由生成元对自由群代数进行局部化，即自由群的 Fox 代数。因此，链接模（link modules），即 Sato 模，自然成为 Fox 代数上的模，从而证明了唯一性，并诱导出因式分解与合成链之间的双射对应。这是建立域系数（或更一般地，主理想整环系数）的自由代数或自由群群代数上矩阵结构理论的最初一步。
英文摘要：
We discuss interrelations between: Cohn localizations of full square matrices; a Leavitt localization of a row; and the Jacobson quasi-inverses of quasi-regular elements. The latter Jacobson localizations appear naturally and easily in rings which are Hausdorff topological spaces with respect to an ideal topology, pointing out also a connection to specific Gabriel localizations. As a main result and an application we develop a factorization theory for free polynomials with non-zero augmentation over a field. This is inspired by a factorization theory given in the joint work with Mantese for polynomials with constant in non-commutative variables. The basic tool of this research is the localization of a free group algebra by a row of free generators, that is, the Fox algebra of a free group. Hence link modules, that is, Sato modules become naturally modules over Fox algebras, proving a uniqueness and inducing a bijective correspondence between factorizations and composition chains. This is a very first step in a structure theory of matrices over either free algebras or group algebras of free groups with coefficients in a field, or more generally in a principal ideal domain.
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。
