The Fox algebra, Localization and factorizations of free polynomials

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.