Non-commutative factorizations and finite-dimensional representations of free algebras

A very first step to develop non-commutative algebraic geometry is the arithmetic of polynomials in non-commuting variables over a commutative field, that is, the study of elements in free associative algebras. This investigation is presented as a natural extension of the classical theory in one variable by using Leavitt algebras, which are localizations of free algebras with respect to the Gabriel topology defined by an ideal of codimension 1. In particular to any polynomial in n non-commuting variables with non-zero constant term we associate a finite-dimensional module over the free algebra of rank n, which turns out to be simple if and only if the polynomial is irreducible. This approach leads to new insights in the study of the factorization of polynomials into irreducible ones and other related topics, such as an algorithm to divide polynomials or to compute the greatest common divisor between them, or a description of similar polynomials. The case of polynomials with zero constant term reduces to an open question whether the intersection of nonunital subalgebras of codimension 1 in a free associative algebra is trivial, that is, 0.