Noncommutative $n$-elementary rings (in Ukrainian)

B.V.Zabavskyi; O.M.Romaniv

In this paper it is proved that any right Bezout ring of stable range $n$ is an $(n+1)$-elementary ring. It is shown, that over any right Hermite ring any unimodular row of length 3 is complemented to an elementary matrix.

1. Ara P., Goodearl K., O'Meara K.C., Pardo E. Diagonalization of matrices over regular rings // Linear Algebra and Appl. -- 1987. -- V.265. -- P.147-163.

2. Menal P., Moncasi J. On regular rings with stable range 2 // J. Pure Appl. Algebra. --1982. -- V.24. -- P.25-40.

3. Zabavsky B.V. Diagonalizability theorem for matrices over rings with finite stable range // Alg. and Discr. Math. --2005. --№1. --151-165.

4. Henriksen M. On a class of regular rings that are elementary divisor rings // Arch. Math. -- 1973. -- V.24, №2. -- P.133-141.

5. Levy L.S. Sometimes only square matrices can be diagonalized // Proc. Amer. Math. Soc. -- 1975. -- V.52. -- P.18-22.

6. Chen H. Generalized stable exchange rings // South Asian Bull. Math. -- 2000. -- V.24 -- P.19-24.

7. Zabavsky B.V. Diagonalization of matrices // Математичні студії. --2005. -- Т.23, №1. -- C.3-10.

8. Zabavsky B.V. Diagonalization of matrices over ring with finite stable rank // Вісник Львівського університету. -- 2003. -- Вип.61. -- С.206-210.

9. Bougaut B. Anneaux Quasi-Euclidiens // These de docteur troisieme cycle. -- 1976. -- V.67.

10. Cooke G.A. A weakening of the euclidean property for integral domains and applications to algebraic number theory. I. // J. fur die Reine and Angw. Math. --1976. -- V.282. -- P. 133-156.

11. Zabavsky B., Romaniv O. Noncommutative rings with elementary reduction of matrices // Вопросы алгебры (Гомель). -- 1999. -- Вып.14. -- С.79-85.

12. Henriksen M. Some remarks about elementary divisor rings // Michigan Math. J. -1955/56. -- V.3. -- P.159-163.

13. Vaserstein L.N. The stable rank of rings and dimensionality of to\-po\-lo\-gical spaces // Functional Anal. Appl. -- 1971. -- V.5. -- P.102-110.

	Noncommutative $n$-elementary rings (in Ukrainian)
Author	B.V.Zabavskyi, O.M.Romaniv Ivan Franko National University of Lviv
Abstract	In this paper it is proved that any right Bezout ring of stable range $n$ is an $(n+1)$-elementary ring. It is shown, that over any right Hermite ring any unimodular row of length 3 is complemented to an elementary matrix.
Keywords	Bezout ring, stable range, Hermite ring
DOI	doi:10.30970/ms.27.1.95-99
Reference	1. Ara P., Goodearl K., O'Meara K.C., Pardo E. Diagonalization of matrices over regular rings // Linear Algebra and Appl. -- 1987. -- V.265. -- P.147-163. 2. Menal P., Moncasi J. On regular rings with stable range 2 // J. Pure Appl. Algebra. --1982. -- V.24. -- P.25-40. 3. Zabavsky B.V. Diagonalizability theorem for matrices over rings with finite stable range // Alg. and Discr. Math. --2005. --№1. --151-165. 4. Henriksen M. On a class of regular rings that are elementary divisor rings // Arch. Math. -- 1973. -- V.24, №2. -- P.133-141. 5. Levy L.S. Sometimes only square matrices can be diagonalized // Proc. Amer. Math. Soc. -- 1975. -- V.52. -- P.18-22. 6. Chen H. Generalized stable exchange rings // South Asian Bull. Math. -- 2000. -- V.24 -- P.19-24. 7. Zabavsky B.V. Diagonalization of matrices // Математичні студії. --2005. -- Т.23, №1. -- C.3-10. 8. Zabavsky B.V. Diagonalization of matrices over ring with finite stable rank // Вісник Львівського університету. -- 2003. -- Вип.61. -- С.206-210. 9. Bougaut B. Anneaux Quasi-Euclidiens // These de docteur troisieme cycle. -- 1976. -- V.67. 10. Cooke G.A. A weakening of the euclidean property for integral domains and applications to algebraic number theory. I. // J. fur die Reine and Angw. Math. --1976. -- V.282. -- P. 133-156. 11. Zabavsky B., Romaniv O. Noncommutative rings with elementary reduction of matrices // Вопросы алгебры (Гомель). -- 1999. -- Вып.14. -- С.79-85. 12. Henriksen M. Some remarks about elementary divisor rings // Michigan Math. J. -1955/56. -- V.3. -- P.159-163. 13. Vaserstein L.N. The stable rank of rings and dimensionality of to\-po\-lo\-gical spaces // Functional Anal. Appl. -- 1971. -- V.5. -- P.102-110.
Pages	95-99
Volume	27
Issue	1
Year	2007
Journal	Matematychni Studii
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