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 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 Bezout 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 Prufer 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() verify an equivalent formulation of (C).
Factorizations into idempotent factors of matrices over Prüfer domains
Cossu, Laura;
2019-01-01
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 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 Bezout 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 Prufer 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() verify an equivalent formulation of (C).
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