Diagonal reduction of matrices over finite stable range rings |
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Author |
b_zabava@ukr.net
Ivan Franko National University of Lviv
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Abstract |
The aim of this review is to present the results of the participants of the scientific seminar
Problems of elementary divisor rings" concerning the Bezout rings of finite stable range.
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Keywords |
diagonal reduction; finite stable range; Bezout ring; exchange ring; Dirichlet ring; ring of neat
range 1
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Reference |
1. Kaplansky I. Elementary divisors and modules// Trans. Amer. Math. Soc. – 1949. – P. 464–491.
2. Gillman L., Henriksen M. Some remarks about elementary divisor rings// Trans. Amer. Math. Soc. – 1956. – V.82. – P. 362–365. 3. Larsen M., Lewis W., Shores T. Elementary divisor rings and finitely presented modules// Trans. Amer. Math. Soc. – 1974. – V.187. – P. 231–248. 4. Henriksen M. Some remarks on elementary divisor rings// Michigan Math. J. – 1955–1956. – V.3. – P. 159–163. 5. Cohn P.M. Free rings and their relations. – Academic Press, London, New York, 1971. – 422 p. 6. Henriksen M. On a class of regular rings that are elementary divisor rings// Arch. Math. – 1973. – V.24, ¹2. – P. 133–141. 7. Zabavsky B.V. Simple elementary divisor rings// Mat. Stud. – 2004. – V.22, ¹2 – P. 129–133. (in Russian) 8. Zabavsky B.V., Komarnytskii M.Ya. Distributive elementary divisor domains// Ukr. Math. J. – 1990. – V.42, ¹7. – P. 1000–1004. (in Ukrainian) 9. Tuganbaev A.A. Rings of elementary divisors and distributive rings// Russian Math. Surveys. – 1991. – V.46, ¹6. – P. 219–220. (in Russian) 10. Bass H. K-theory and stable algebra// Inst. Hautes Etudes. Sci. Publ. Math. – 1964. – V.22. – P. 485–544. 11. Zabavsky B.V. Reduction of matrices over Bezout rings of stable rank not higher than 2// Ukr. Math. J. – 2003. – V.55, ¹4. – P. 550–554. (in Ukrainian) 12. Couchot F. The lambda-dimension of commutative arithmetic ring// Commun. in Algebra. – 2004. – V.31, ¹7. – P. 1–14. 13. Shores T., Wiegand R. Decomposition of modules and matrices// Bull. Amer. Math. Soc. – 1973. – V.79, ¹6. – P. 1277–1280. 14. Zabavsky B.V., Domsha O.V. Kazimirsky’s rings// Mat. Stud. – 2010. – V.34, ¹1. – P. 75–79. (in Ukrainian) 15. Helmer O. The elementary divisor for certain rings without chain conditions// Bull. Amer. Math. Soc. – 1943. – V.49, ¹2. – P. 225–236. 16. Zabavsky B.V., Bilyavs’ka S.I. Stable rank of adequate ring// Mat. Stud. – 2010. – V.33, ¹2. – P. 212–214. (in Ukrainian) 17. McGovern W. Personal communication. – 2011. 18. Nicholson W.K., Sanchez Campos E. Rings with the dual of the isomorphism theorem// J. Algebra. – 2008. – V.212. – P. 340-348. 19. Zabavsky B.V., Bilyavs’ka S.I. Decomposition of finitely generated projective modules over Bezout ring// Mat. Stud. – 2013. – V.40, ¹1. – P. 104–107. 20. 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. 21. Zabavsky B.V. Rings over which every matrix admits the diagonal reduction by elementary transformations// Mat. Stud. – 1997. – V.8, ¹2. – P. 136–139. (in Ukrainian) 22. Zabavsky B.V. Elementary reduction of matrices over adequate domain// Mat. Stud. – 2002. – V.17, ¹2. – P. 115–116. 23. Romaniv O.M. Elementary reduction of matrices over Bezout ring witn n-fold stable range 1// Applied Problems of Mechanics and Mathematics. – 2012. – V.10. – P. 77–79. (in Ukrainian) 24. Zabavsky B.V., Romaniv O.M. Elementary reduction of matrices over commutative Bezout ring witn n-fold stable range 2// Applied Problems of Mechanics and Mathematics. – 2013. – V.11. – P. 141–144. (in Ukrainian) 25. Zabavsky B.V. Diagonal reduction of matrices over rings. – Mathematical Studies, Monograph Series, V.XVI, VNTL Publishers, 2012, Lviv. – 251 p. 26. Gillman L., Henriksen. M. Rings of continuous function in which every finitely generated ideal is principal// Trans. Amer. Math. Soc. – 1956. – V.82. – P. 366–394. 27. Tuganbaev A.A. Ring Theory. Arithmetical Modules and Rings. – M.: MCCME, 2009. – 422 p. (in Russian) 28. Zabavsky B.V. A sharp Bezout domain is an elementary divisor ring// Ukr. Math. J. – 2014. – V.66, ¹2. – P. 284–288. (in Ukrainian) 29. Roitman M. The Kaplansky coudition and rings of almost stable range 1// Trans. Amer. Math. Soc. – 2013. – V.141, ¹9. – P. 3013–3019. 30. Shchedryk V.P. Commutative domains of elementary divisors and some properties of their elements// Ukr. Math. J. – 2012. – V.64, ¹1. – P. 140–155. (in Ukrainian) |
Pages |
101-108
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Volume |
41
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Issue |
1
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Year |
2014
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Journal |
Matematychni Studii
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Full text of paper | |
Table of content of issue |