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 (ID_n): “every singular n × n matrix over R is a product of idempotent matrices”. Significant results in [18, 21] and [5] motivated a natural conjecture, proposed by Salce and Zanardo [22]: (C) “an integral domain R satisfying (ID₂) is necessarily a Bézout domain”. Unique factorization domains, projective-free domains and PRINC domains (cf. [22]) verify the conjecture. We prove that an integral domain R satisfying (ID₂) must be a Prüfer domain in which every invertible 2 × 2 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($\mathbb{Z}$) verify an equivalent formulation of (C).

Keywords:

• Factorization of matrices
• idempotent matrices
• elementary matrices
• Prüfer domains

2010 Mathematics Subject Classification:

• Primary: 15A23; Secondary: 13F05
• 14H50
• 13F20

Acknowledgements

The authors would like to thank the referee for valuable comments and suggestions that helped to considerably improve the first version of this paper.