Alahmadi, Adel; Jain, S. K.; Lam, T. Y.; Leroy, A. Euclidean pairs and quasi-Euclidean rings. (English) Zbl 1318.16036 J. Algebra 406, 154-170 (2014). The authors study the interplay between quasi-Euclidean properties and matrix properties (such as diagonalization of rectangular matrices and factorization into products of idempotent matrices) over noncommutative unital rings. Their fundamental concept is that of a right Euclidean pair for a ring \(R\), meaning an ordered pair \((a,b)\in R^2\) which has a terminating sequence of divisions with remainders of the form \(a=bq_1+r_1\), \(b=r_1q_2+r_2\), \(r_1=r_2q_3+r_3\),…, \(r_{n-1}=r_nq_{n+1}+r_{n+1}\) for some \(q_i,r_i \in R\) with \(r_{n+1}=0\). The ring \(R\) is called right quasi-Euclidean if all ordered pairs in \(R^2\) are right Euclidean. These properties are related to the possibilities that \(R\) is a right K-Hermite ring (for each \((a,b)\in R^2\), there exist \(r\in R\) and \(Q\in GL_2(R)\) such that \((a,b)=(r,0)Q\)), a \(GE_n\) ring (meaning that \(GL_n(R)\) is generated by elementary matrices), or a right Bezout ring (every finitely generated right ideal is principal). The main results include the following: (I) A ring \(R\) is right quasi-Euclidean if and only if it is a \(GE_2\)-ring and a right K-Hermite ring, in which case it is a \(GE_n\)-ring for all \(n\). (II) An integral domain \(R\) is right quasi-Euclidean if and only if \(R\) is a \(GE_2\)-ring, every finitely generated projective right \(R\)-module is free of unique rank, and every matrix \(\left(\begin{smallmatrix} a&b\\ 0&0\end{smallmatrix}\right)\in M_2(R)\) is a product of idempotent matrices. (III) Every matrix ring over a right quasi-Euclidean ring is right quasi-Euclidean. (IV) A left quasi-Euclidean ring is right quasi-Euclidean if and only if it is a right Bezout ring, and a left quasi-Euclidean domain is right quasi-Euclidean if and only if it is a right Ore domain. (V) If \(R\) is a right and left quasi-Euclidean domain, then every matrix in \(M_n(R)\) whose left (or right) annihilator in \(M_n(R)\) is nonzero is a product of idempotent matrices. This generalizes the result for commutative Euclidean domains due to T. J. Laffey [Linear Multilinear Algebra 14, 309-314 (1983; Zbl 0526.15008)]. The authors also prove that every unit-regular ring is right and left quasi-Euclidean. In contrast, a famous example of G. Bergman appearing in [D. Handelman, J. Algebra 48, 1-16 (1977; Zbl 0363.16009)] is shown to be a right and left quasi-Euclidean, von Neumann regular ring which is not even Dedekind finite (directly finite). Reviewer: Kenneth R. Goodearl (Santa Barbara) Cited in 1 ReviewCited in 15 Documents MSC: 16U30 Divisibility, noncommutative UFDs 13F07 Euclidean rings and generalizations 16U80 Generalizations of commutativity (associative rings and algebras) 16S50 Endomorphism rings; matrix rings 16E50 von Neumann regular rings and generalizations (associative algebraic aspects) 15A23 Factorization of matrices 15A30 Algebraic systems of matrices Keywords:noncommutative Euclidean rings; quasi-Euclidean rings; Hermite rings; GE-rings; Bézout rings; idempotent matrices; unit-regular rings; von Neumann regular rings Citations:Zbl 0526.15008; Zbl 0363.16009 PDFBibTeX XMLCite \textit{A. Alahmadi} et al., J. Algebra 406, 154--170 (2014; Zbl 1318.16036) Full Text: DOI References: [1] Alahmadi, A.; Jain, S. K.; Leroy, A., Decomposition of singular matrices into idempotents, Linear Multilinear Algebra, 62, 1, 13-27 (January 2014) [2] Amitsur, S. A., Remarks on principal ideal rings, Osaka Math. J., 15, 59-69 (1963) · Zbl 0113.25904 [3] Bhaskara Rao, K. P.S., Products of idempotent matrices, Linear Algebra Appl., 430, 2690-2695 (2009) · Zbl 1165.15016 [4] Cohn, P. M., On the structure of the \(GL_2\) of a ring, IHES Publ. Math., 30, 5-53 (1966) [5] Cohn, P. M., Free Ideal Rings and Localization in General Rings, New Math. Monogr., vol. 3 (2006), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1114.16001 [6] Erdos, J. A., On products of idempotent matrices, Glasg. Math. J., 8, 118-122 (1967) · Zbl 0157.07101 [7] Fountain, J., Products of idempotent integer matrices, Math. Proc. Cambridge Philos. Soc., 110, 431-441 (1991) · Zbl 0751.20049 [8] Glivický, P.; Šaroch, J., Quasi-Euclidean subrings of \(Q [x]\), Comm. Algebra, 41, 11, 4267-4277 (November 2013) [9] Goodearl, K. R., Von Neumann Regular Rings (1991), Krieger Publishing Company: Krieger Publishing Company Malabar, Florida · Zbl 0749.16001 [10] Handelman, D., Perspectivity and cancellation in regular rings, J. Algebra, 48, 1-16 (1977) · Zbl 0363.16009 [11] Howie, J. M., The subsemigroup generated by the idempotents of a full transformation semigroup, J. Lond. Math. Soc., 41, 707-716 (1966) · Zbl 0146.02903 [12] Kaplansky, I., Elementary divisors and modules, Trans. Amer. Math. Soc., 66, 464-491 (1949) · Zbl 0036.01903 [13] Laffey, T. J., Products of idempotent matrices, Linear Multilinear Algebra, 14, 309-314 (1983) · Zbl 0526.15008 [14] Lam, T. Y., A First Course in Noncommutative Rings, Grad. Texts in Math., vol. 131 (2001), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0980.16001 [15] Lam, T. Y., Lectures on Modules and Rings, Grad. Texts in Math., vol. 189 (1999), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0911.16001 [16] Lam, T. Y., Serre’s Problem on Projective Modules, Monogr. Math. (2006), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 1101.13001 [17] Leutbecher, A., Euklidischer Algorithmus und die Gruppe \(GL_2\), Math. Ann., 231, 269-285 (1978) · Zbl 0367.16001 [18] McCoy, N., Rings and Ideals, Carus Math. Monogr. (1948), Math. Assoc. America [19] Menal, P.; Moncasi, J., On regular rings with stable range 2, J. Pure Appl. Algebra, 24, 25-40 (1982) · Zbl 0484.16006 [20] O’Meara, O. T., On the finite generation of linear groups over Hasse domains, J. Reine Angew. Math., 217, 79-128 (1964) · Zbl 0128.25502 [21] Ruitenburg, W., Products of idempotent matrices over Hermite domains, Semigroup Forum, 46, 371-378 (1993) · Zbl 0786.15017 [22] Stepanov, A. V., A ring of finite stable rank is not necessarily finite in the sense of Dedekind, Sov. Math. Dokl., 36, 301-304 (1988) · Zbl 0651.20048 [23] Zabavsky, B. V., Reduction of matrices over Bezout rings with stable rank not higher than 2, Ukrainian Math. J., 55, 550-554 (2003) · Zbl 1038.16008 [24] Zabavsky, B. V., Diagonalizability theorems for matrices over rings with finite stable range, Algebra Discrete Math., 1, 151-165 (2005) · Zbl 1091.15014 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.