Periodic Reporting for period 1 - FacT-in-MaRs (Factorization Theory in Matrix Rings)
Reporting period: 2021-08-02 to 2023-08-01
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.
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.
Combining the language of preorders and monoids, we have developed a new concept of factorization that extends the classical factorization theory to the study of decompositions into "arbitrary" factors in monoids that are not necessarily commutative or cancellative. For this new "generalized" factorization theory, we have found results related to both the existence and non-uniqueness of the decompositions. In particular, we have provided sufficient conditions for the existence of factorizations and others for the finiteness of their corresponding arithmetic invariants. We have applied these new tools to the analysis of matrix factorization into idempotents, offering alternative proofs of known results and demonstrating original ones. In addition, the applications of this new theory to the classical context have allowed us to generalize results already known for cancellative or "quasi" cancellative monoids to highly non-cancellative settings. The results obtained within FacT-in-MaRs are compiled in three articles, two of which have already been accepted for publication in international journals. These same results have been presented at numerous conferences and invited seminars.
FacT-in-MaRs has connected for the first time two previously unrelated areas of ring theory: the theory of factorization and the problem of factorizing singular square matrices into idempotents. This has led to the development of a generalized theory of factorization with far-reaching applications beyond what has been explored so far. FacT-in-MaRs has laid the groundwork for at least two new research directions: on one hand, further development of the theoretical framework, and on the other hand, the application of the theory itself to specific case studies. There is still much work to be done, especially in the area of matrix factorizations. In summary, FacT-in-MaRs has the potential to have a significant impact on the world of ring and semigroup theory.