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arXiv · 1810.00639

Factorizations into idempotent factors of matrices over Prüfer domains

Laura Cossu·Paolo Zanardo
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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).

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Laura Cossu, Paolo Zanardo. 2018-10-01. Factorizations into idempotent factors of matrices over Prüfer domains. https://doi.org/10.1080/00927872.2018.1523419

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