FacT-in-MaRs aims to study the factorization of square matrices into products of idempotent matrices from the perspective of factorization theory, specifically focusing on the systematic analysis of the non-uniqueness of such decomposition. Characterizing domains (i.e. rings without non-zero zero divisors) R for which all singular matrices with entries in R can be expressed as products of idempotent matrices is indeed a classic open problem in ring theory. This problem is closely tied to the elementary generation of the special linear group of R and the existence of Euclidean-type algorithms.
Most of the classical theory of factorization, originated in the Sixties in the framework of the algebraic number theory, has been developed to study factorizations into atoms (i.e. non-units that cannot be decomposed into the product of two non-units) of non-unit elements of a commutative and cancellative monoid (i.e. a commutative monoid in which ax=bx only if a=b). However, when departing from commutativity and cancellativity, the machinery of the classical theory needs to be substantially reframed.
In fact, the extension of the theory to a non-commutative setting works nicely if there exists a "transfer morphism'' from the monoid under exam to a commutative and [unit-]cancellative monoid, otherwise the theory shows some "gaps'': some ``nice'' and ``small'' monoids do not admit factorization into atoms even if they should morally do; the classical invariants associated with atomic factorizations (e.g. their lengths) blow up in a predictable way and lose most of their significance. Analogous issues appear in highly non-cancellative (even commutative) monoids, e.g. in the presence of non-trivial idempotents or in rings with non-zero zero divisors.
Since we are interested in studying idempotent (and therefore non-atomic) factorizations in non-commutative monoids of square matrices, we need to define a proper notion of factorization, along with its arithmetical invariants, to address the gaps mentioned above.
The objectives of FacT-in-MaRs can be summarized in this way:
(O1) Define a new concept of factorization in which arithmetical invariants are not subject to predictable blow-up phenomena, even when the factors are idempotent elements.
(O2) Study the non-uniqueness of the aforementioned factorization.
(O3) Exploring the problem of idempotent factorization in matrix rings from the perspective of the newly introduced "generalized" factorization theory.
