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

TL;DR

This paper investigates conditions under which matrices over certain integral domains can be factored into idempotent matrices, proving that domains satisfying a specific property are Pr"ufer domains with particular matrix factorization characteristics.

Contribution

It proves that domains satisfying (ID2) are Pr"ufer domains with elementary matrix factorizations and confirms the conjecture for several classes of rings.

Findings

Domains satisfying (ID2) are Pr"ufer domains with elementary matrix factorizations.

Several classes of rings, including coordinate rings of plane curves and integer-valued polynomials, satisfy the conjecture.

The paper characterizes the structure of domains where singular matrices decompose into idempotent factors.

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).