Leírás
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 that 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. The basic tool is the algebra of Fox derivatives 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 the 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.
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