It is known that a simple Bézout domain is a domain of elementary divisors if and only if it is 2-simple. We prove that, over a 2-simple Ore domain of stable rank 1, an arbitrary matrix that is not a divisor of zero is equivalent to a canonical diagonal matrix.
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References
H. J. S. Smith, “On systems of linear indeterminate equations and congruences,” Phil. Trans. Roy. Soc. London, 151, No. 2, 293–326 (1861).
L. E. Dickson, Algebras and Their Arithmetics, Chicago University, Chicago (1923).
J. H. M. Wedderburn, “Non-commutative domains of integrity,” J. Reine Angew. Math., 167, No. 1, 129–141 (1932).
B. L. van der Warden, Moderne Algebra, Springer, Berlin (1930).
N. Jacobson, “Pseudo-linear transformation,” Ann. Math., No. 38, 484–507 (1937).
O. Teichmuller, “Der Elementrateilsatz für neichtkommutative Ringe,” Abh. Preuss. Acad. Wiss. Phys.-Math. K 1, 169–177 (1937).
K. Asano, Neichtkommutative Hauptidealringe, Hermann, Paris (1938).
I. Kaplansky, “Elementary divisors and modules,” Trans. Amer. Math. Soc., 66, 464–491 (1949).
L. Gerstein, “A local approach to matrix equivalence,” Linear Algebra Appl., 16, 221–232 (1977).
L. Gillman and M. Henricsen, “Some remarks about elementary divisor rings,” Trans. Amer. Math. Soc., 82, 362–365 (1956).
M. Henricsen and M. Jerison, “The space of minimal primes of commutative rings,” Trans. Amer. Math. Soc., 115, 110–130 (1965).
T. Y. Lam, A Crash Course on Stable Range, Cancellation, Substitution and Exchange, University of California, Berkeley (1972).
B. V. Zabavskii, “On noncommutative rings with elementary divisors,” Ukr. Mat. Zh., 42, No. 6, 847–850 (1990).
B. V. Zabavsky, “Diagonalization of matrices,” Mat. Stud., 23, No. 1, 3–10 (2005).
P. M. Cohn, “On the structure of the GL 2 of a ring,” Publ. Math. I.H.E.S., No. 30, 5–59 (1966).
P. M. Cohn, “Rings of a transfinite weak algorithm,” Bull. London Math. Soc., No. 1, 55–59 (1969).
B. V. Zabavskii, “Simple elementary divisor rings,” Mat. Stud., 22, No. 2, 129–133 (2004).
B. V. Zabavsky, “Diagonalizability theorem for matrices over ring with finite stable range,” Algebra Discrete Math., No. 1, 134–148 (2005).
J. Olszewski, “On ideals of product of rings,” Demonstr. Math., 27, No. 1, 1–7 (1994).
L. N. Vaserstein, “The stable rank of rings and dimensionality of topological spaces,” Funct. Anal. Appl., No. 5, 102–110 (1971).
P. M. Cohn, Free Rings and Their Relations, Academic Press, London (1971).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 10, pp. 1436–1440, October, 2010.
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Domsha, O.V., Zabavs’kyi, B.V. 2-Simple ore domains of stable rank 1. Ukr Math J 62, 1666–1672 (2011). https://doi.org/10.1007/s11253-011-0458-3
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DOI: https://doi.org/10.1007/s11253-011-0458-3