Factorizations into idempotent factors of matrices over Prüfer 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 Bézout domain, motivated a natural conjecture, proposed by Salce and Zanardo in 2014: (C) "an integral domain R satisfying (ID2) is necessarily a Bézout 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 Prüfer 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).

References in corpus (1)

• Decomposition of Singular Matrices into Idempotents

Cited by in corpus (5)

• Abstract Factorization Theorems with Applications to Idempotent Factorizations
• Idempotent factorizations of singular $2\times 2$ matrices over quadratic integer rings
• Minimal Prüfer-Dress rings and products of idempotent matrices
• Effectively bounded idempotent generation of certain $2\times 2$ singular matrices by idempotent matrices over real quadratic number rings
• PRINC domains and comaximal factorization domains