Periodic Reporting for period 1 - FacT-in-MaRs (Factorization Theory in Matrix Rings)
Période du rapport: 2021-08-02 au 2023-08-01
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.