Abstract
A classical problem that goes back to the 1960’s, is to characterize the integral domains R satisfying the property (IDn): “every singular n × n matrix over R is a product of idempotent matrices”. Significant results in [Citation18, Citation21] and [Citation5] motivated a natural conjecture, proposed by Salce and Zanardo [Citation22]: (C) “an integral domain R satisfying (ID2) is necessarily a Bézout domain”. Unique factorization domains, projective-free domains and PRINC domains (cf. [Citation22]) 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() verify an equivalent formulation of (C).
2010 Mathematics Subject Classification:
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.