Factorizations into idempotent factors of matrices over Pr\"ufer domains

Abstract

A classical problem, that goes back to the 1960's, is to characterize the integral domains R satisfying the property (IDn): "every singular nxn matrix over R is a product of idempotent matrices". Significant results, which describe this property in the class of B\'ezout domain, motivated a natural conjecture, proposed by Salce and Zanardo in 2014: (C) "an integral domain R satisfying (ID2) is necessarily a B\'ezout domain". Unique factorization domains, projective-free domains and PRINC domains verify the conjecture. We prove that an integral domain R satisfying (ID2) must be a Pr\"ufer domain in which every invertible 2x2 matrix is a product of elementary matrices. Then we show that a large class of coordinate rings of plane curves and the ring of integer-valued polynomials Int(Z) verify an equivalent formulation of (C).