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Decompositions into products of idempotents. (English) Zbl 1326.15019

Summary: The purpose of this note is two-fold: (1) to study when quasi-Euclidean rings, regular rings and regular separative rings have the property \((*)\) that each right (left) singular element is a product of idempotents, and (2) to consider the question: “when is a singular nonnegative square matrix a product of nonnegative idempotent matrices?” The importance of the class of quasi-Euclidean rings in connection with the property \((*)\) is given by the first three authors and T. Y. Lam J. Algebra 406, 154–170 (2014; Zbl 1318.16036)], where it is shown that every singular matrix over a right and left quasi-Euclidean domain is a product of idempotents, generalizing the results of J. A. Erdos [Glasg. Math. J. 8, 118–122 (1967; Zbl 0157.07101)] for matrices over fields and that of T. J. Laffey [Linear Multilinear Algebra 14, 309–314 (1983; Zbl 0526.15008)] for matrices over commutative Euclidean domains. We have shown in this paper that quasi-Euclidean rings appear among many interesting classes of rings and hence they are in abundance. We analyze the properties of triangular matrix rings and upper triangular matrices with respect to the decomposition into product of idempotents and show, in particular, that nonnegative nilpotent matrices are products of nonnegative idempotent matrices. We study as to when each singular matrix is a product of idempotents in special classes of rings. Regarding the second question for nonnegative matrices, bounds are obtained for a rank one nonnegative matrix to be a product of two idempotent matrices. It is shown that every nonnegative matrix of rank one is a product of three nonnegative idempotent matrices. For matrices of higher orders, we show that some power of a group monotone matrix is a product of idempotent matrices.

MSC:

15A23 Factorization of matrices
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A30 Algebraic systems of matrices
16S50 Endomorphism rings; matrix rings
16U30 Divisibility, noncommutative UFDs
15B48 Positive matrices and their generalizations; cones of matrices
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