Fractionally regular Bezout rings

B.V.Zabavsky

It is proved that a fractionally regular Bezout ring has stable range 3. A PM-ring which is a fractionally regular Bezout ring is an elementary divisor ring. It is proved that a fractionally regular Bezout ring of stable rang 2 is an elementary divisor ring. A Bezout ring of stable range 2 with right Krull dimension is an elementary divisor ring. A Bezout ring of stable range 2 with Noetherian spectrum is an elementary divisor ring.

1. Vamos P. The decomposition of finitely generated modules and fractionally self-injective rings // J.~London Math. Soc. - 1977. - V.6. - P.209-220.

2. Facctini A., Faith O. FP-ingective quotient rings and elementary divisor rings. Commut. ring theory // Proc. Int. conf. - 1996. - V.185. - P.293-302.

3. Ohm. J. Pendleton R. A note on rings with Noetherian spectrum // Duke Math. J. - 1968. - V.35. - P.631-639.

4. Matlis E. The minimal spectrum of a reduced ring // Illinois J. Math. - 1983. - V.27, №3. - P.353-391.

5. Kaplansky I. Elementary divisors and modules // Trans. Amer. Math. Soc. - 1959. - V.66. - P.464-491.

6. Gillman L., Henriksen M. Some remarks about elementary divisor rings // Trans. Amer. Math. Soc. - 1956. - V.82. - P.362-365.

7. Vaserstein L. The stable rank of rings and dimensionality of topological spaces // Funtional. Anal. Appl. - 1971. - V.5. - P.102-110.

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

9. Henriksen M., Jerison M. The space of minimal prime ideals of a commutative rings // Trans. Amer. Math. Soc. - 1965. - V.115. - P.110-130.

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

11. Zabavsky B. Reduction of matrices over Bezout rings of stable rank not higher that 2 // Ukr. Math. J. - 2003. - V.55, №4. - P.550-554.

12. Arapovic M. Characterizations of the 0-dimensional rings // Glas. Math. - 1983. - V.18. - P.39-49.

13. De Marco G., Orsati A. Commutative ring in which every prime ideal is contained in a unique maximal ideal // Proc. Amer. Math. Soc. - 1971. - V.30. - P.459-466.

14. Couchot F. Indecomposable modules and Gelfand rings // Commun. in Alg. - 2007. V.35. - P.231-241.

15. Gillman L., Henriksen M. Rings of continuous functions in which every finitely ideals is principal // Trans. Amer. Math. Soc. - 1956. - V.82. - P.366-391.

16. Shores T. Modules over semihereditary Bezout rings // Proc. Amer. Math. Soc. - 1974. - V.46, №2. - P.211-213.

17. Henriksen M. Some remarks on elementary divisor rings // Michigan. Math. J. - 1955/1956. - V.3 - P.159-163.

18. Larsen M., Lewis W., Shores T. Elementary divisor rings and finitely presented modules// Trans. Amer. Math. Soc. - 1974. - V.187. - V.231-248.

19. Puninski G., Rothmaler P. When every finitely generated flat modules is projective // J. Algebra. - 2004. - V.277. - P.542-558.

20. Kaplansky I. Commutative Rings. - Chicago and London: The University of Chicago Press, 1974.

	Fractionally regular Bezout rings
Author	B.V.Zabavsky b_zabava@ukr.net Ivan Franko National University of Lviv,Universytetska 1, 79000, Ukraine
Abstract	It is proved that a fractionally regular Bezout ring has stable range 3. A PM-ring which is a fractionally regular Bezout ring is an elementary divisor ring. It is proved that a fractionally regular Bezout ring of stable rang 2 is an elementary divisor ring. A Bezout ring of stable range 2 with right Krull dimension is an elementary divisor ring. A Bezout ring of stable range 2 with Noetherian spectrum is an elementary divisor ring.
Keywords	fractionally regular Bezout ring, stable range, elementary divisor ring, Krull dimension, Noetherian spectrum
DOI	doi:10.30970/ms.32.1.76-80
Reference	1. Vamos P. The decomposition of finitely generated modules and fractionally self-injective rings // J.~London Math. Soc. - 1977. - V.6. - P.209-220. 2. Facctini A., Faith O. FP-ingective quotient rings and elementary divisor rings. Commut. ring theory // Proc. Int. conf. - 1996. - V.185. - P.293-302. 3. Ohm. J. Pendleton R. A note on rings with Noetherian spectrum // Duke Math. J. - 1968. - V.35. - P.631-639. 4. Matlis E. The minimal spectrum of a reduced ring // Illinois J. Math. - 1983. - V.27, №3. - P.353-391. 5. Kaplansky I. Elementary divisors and modules // Trans. Amer. Math. Soc. - 1959. - V.66. - P.464-491. 6. Gillman L., Henriksen M. Some remarks about elementary divisor rings // Trans. Amer. Math. Soc. - 1956. - V.82. - P.362-365. 7. Vaserstein L. The stable rank of rings and dimensionality of topological spaces // Funtional. Anal. Appl. - 1971. - V.5. - P.102-110. 8. Henriksen M. On a class of regular rings that are elementary divisor rings // Arch. Math. - 1973. - V.24, №2. - P.133-141. 9. Henriksen M., Jerison M. The space of minimal prime ideals of a commutative rings // Trans. Amer. Math. Soc. - 1965. - V.115. - P.110-130. 10. Zabavsky B. Diagonalizability theorem for matrices over rings with finite stable range // Alg. Discr. Math. - 2005. - №1. - P.151-165. 11. Zabavsky B. Reduction of matrices over Bezout rings of stable rank not higher that 2 // Ukr. Math. J. - 2003. - V.55, №4. - P.550-554. 12. Arapovic M. Characterizations of the 0-dimensional rings // Glas. Math. - 1983. - V.18. - P.39-49. 13. De Marco G., Orsati A. Commutative ring in which every prime ideal is contained in a unique maximal ideal // Proc. Amer. Math. Soc. - 1971. - V.30. - P.459-466. 14. Couchot F. Indecomposable modules and Gelfand rings // Commun. in Alg. - 2007. V.35. - P.231-241. 15. Gillman L., Henriksen M. Rings of continuous functions in which every finitely ideals is principal // Trans. Amer. Math. Soc. - 1956. - V.82. - P.366-391. 16. Shores T. Modules over semihereditary Bezout rings // Proc. Amer. Math. Soc. - 1974. - V.46, №2. - P.211-213. 17. Henriksen M. Some remarks on elementary divisor rings // Michigan. Math. J. - 1955/1956. - V.3 - P.159-163. 18. Larsen M., Lewis W., Shores T. Elementary divisor rings and finitely presented modules// Trans. Amer. Math. Soc. - 1974. - V.187. - V.231-248. 19. Puninski G., Rothmaler P. When every finitely generated flat modules is projective // J. Algebra. - 2004. - V.277. - P.542-558. 20. Kaplansky I. Commutative Rings. - Chicago and London: The University of Chicago Press, 1974.
Pages	76-80
Volume	32
Issue	1
Year	2009
Journal	Matematychni Studii
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