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Ukrainian Mathematical Journal

, Volume 55, Issue 4, pp 665–670 | Cite as

Reduction of Matrices over Bezout Rings of Stable Rank not Higher than 2

  • B. V. Zabavs'kyi
Article

Abstract

We prove that a commutative Bezout ring is an Hermitian ring if and only if it is a Bezout ring of stable rank 2. It is shown that a noncommutative Bezout ring of stable rank 1 is an Hermitian ring. This implies that a noncommutative semilocal Bezout ring is an Hermitian ring. We prove that the Bezout domain of stable rank 1 with two-element group of units is a ring of elementary divisors if and only if it is a duo-domain.

Keywords

Stable Rank Elementary Divisor Bezout Ring Bezout Domain Hermitian Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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REFERENCES

  1. 1.
    M. Henriksen, “Some remarks about elementary divisor rings,” Mich. Math. J., 3, 159-163 (1955/56).Google Scholar
  2. 2.
    M. Larsen, W. Lewis, and T. Shores, “Elementary divisor rings and finitely presented modules,” Trans. Amer. Math. Soc., 187, No. 1, 231-248 (1974).Google Scholar
  3. 3.
    P. Menal and J. Moncasi, “On regular rings with stable range 2,” J. Pure Appl. Algebra, 24, 25-40 (1982).Google Scholar
  4. 4.
    I. Kaplansky, “Elementary divisors and modules,” Trans. Amer. Math. Soc., 66, No. 2, 464-491 (1949).Google Scholar
  5. 5.
    L. N. Vasershtein, “Stable rank of rings and dimension of topological spaces,” Funkts. Anal., 5, 17-27 (1971).Google Scholar
  6. 6.
    A. I. Gatalevich, “On duo-rings of elementary divisors,” in: Algebra and Topology [in Ukrainian], Lviv (1996), pp. 58-65.Google Scholar
  7. 7.
    J. Olszewski, “On ideals of products of rings,” Demonstr. Math., 1, 1-7 (1994).Google Scholar
  8. 8.
    B. V. Zabavs'kyi and M. Ya. Komarnyts'kyi, “Distributive domains of elementary divisors,” Ukr. Mat. Zh., 42, No. 7, 339-344 (1990).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • B. V. Zabavs'kyi
    • 1
  1. 1.Lviv UniversityLviv

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