PRINC domains and comaximal factorization domains

The notion of a PRINC domain was introduced by Salce and Zanardo [Products of elementary and idempotent matrices over integral domains, Linear Algebra Appl. 452 (2014) 130-152], motivated by the investigation of the products of idempotent matrices with entries in a commutative domain. An integral domain R is a PRINC domain if every two-generated invertible ideal of R is principal. PRINC domains are closely related to the notion of a unique comaximal factorization domain, introduced by McAdam and Swan [Unique comaximal factorization, J. Algebra 276 (2004) 180-192]. In this paper, we prove that there exist large classes of PRINC domains which are not comaximal factorization domains, using diverse kinds of constructions. We also produce PRINC domains that are neither comaximal factorization domains nor projective-free.