Abstract
We introduce a concept of noncommutative (right) 2-Euclidean ring. We prove that a 2-Euclidean ring is a right Hermite ring, a right Bezout ring, and a GE n -ring. It is shown that an arbitrary right unimodular string of length not less than 3 over a right Bezout ring of stable rank possesses an elementary diagonal reduction. We prove that a right Bezout ring of stable rank 1 is a right 2-Euclidean ring.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
B. V. Zabavs’kyi, “Rings over which an arbitrary matrix admits a diagonal reduction by elementary transformations,” Mat. Stud., 8, No.2, 136–139 (1997).
B. V. Zabavs’kyi and O. M. Romaniv, “Commutative 2-Euclidean rings,” Mat. Stud., 15, No.2, 140–144 (2001).
B. Bougaut, Anneaux Quasi-Euclidiens, These de Docteur Troisieme Cycle (1976).
P. M. Cohn, “On the structure of the GL 2 of ring,” I. H. E. S. Publ. Math., 30, 365–413 (1966).
I. Kaplansky, “Elementary divisors and modules,” Trans. Amer. Math. Soc., 66, 464–491 (1949).
P. M. Cohn, Free Rings and Their Relations, Academic Press, London (1971).
R. B. Warfield, “Stable equivalence of matrices and resolutions,” Commun. Algebra, 6, 1811–1828 (1978).
Author information
Authors and Affiliations
Additional information
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1717 – 1721, December, 2004.
Rights and permissions
About this article
Cite this article
Romaniv, O.M. Elementary Reduction of Matrices over Right 2-Euclidean Rings. Ukr Math J 56, 2028–2034 (2004). https://doi.org/10.1007/s11253-005-0167-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-005-0167-x