Title: Products of idempotent matrices, PRINC domains, comaximal factorization domains
Speaker: Paolo Zanardo (University of Padova)
Abstract: A much investigated question, concerned with factorization of matrices over a commutative domain $R$, is whether every $n \times n$ singular matrix is a product of idempotent matrices. If this is the case, we say that $R$ satisfies property $(ID_n)$. Several relevant results on this subject led Salce and Zanardo (2014) to propose the naturalconjecture: (C) An integral domain $R$ satisfying property $(ID_2)$ is necessarily a Bezout domain.
Cossu and Zanardo (2018) investigated property $(ID_2)$ and found classes of rings that satisfy (C), namely, special coordinate rings of smooth algebraic curves, and the ring of integer-valued polynomials.
The notion of PRINC domain was introduced by Salce and Zanardo (2014). They proved that the class of PRINC domains satisfies (C). An integral domain $R$ is a PRINC domain if every two-generated invertible ideal of $R$ is principal.
If $R$ is either factorial or projective-free, then it is PRINC.
PRINC domains are closely related to unique comaximal factorization domains, a notion introduced by McAdam and Swan (2004).
In a joint work with Laura Cossu (2019), we proved that there exist large classes of PRINC domains that are not comaximal factorization domains, using diverse kinds of constructions. We also produced PRINC domains that are neither comaximal factorization domains nor projective-free.
