Factorization Theory in Matrix Rings

The characterization of integral domains R such that every singular matrix over R is a product of idempotent matrices is a classical open problem in ring theory.

Its importance lies in the inter-connections with other big unsolved issues: classify integral domains whose general linear groups are generated by the elementary matrices, and those fulfilling weak versions of the Euclidean algorithm.

The study of idempotent factorizations in matrix rings has gained increasing attention over the years and all the results have highlighted how the decomposition into idempotent factors is far from being unique.The Factorization Theory (FT) is the branch of ring theory that studies nonuniqueness of the representation of non-invertible elements in rings or semigroups as products of generating (irreducible) elements.

Originated in the late 1960s, FT got in the last decade new striking developments (especially in the non-commutative framework) that, however, just barely involved matrix rings.The goal of FacT-in-MaRs is to study the nonuniqueness phenomena of idempotent matrix factorization from the point of view of the FT, thus connecting in an original way two areas of ring theory remained unrelated so far.In the framework of the present action, we aim at advancing the state-of-the-art by:1) defining a new concept of factorization into idempotent (non-irreducible) factors in the non-commutative semigroup of singular matrices over a domain R; 2) studying the nonuniqueness of this factorization in terms of arithmetical invariants (i.e., sets of legths/distances, elasticity);3) exploiting the previous results to provide new approaches to the classical problems on factorizations in matrix rings.The above objectives will be achieved through an innovative combination of classical and recent techniques of the theory of factorization of matrices over integral domains and of the FT, respectively belonging to the background of the applicant and of the Supervisor.