Diagonalization of matrices

B.V.Zabavsky

We consider the question of diagonalizability of matrices of given size over regular rings of finite stable range as well as ``weak'' diagonal reduction of arbitrary regular ring.

1. Smith H.J.S. On systems of linear indeterminate equations and congruences, Philos. Trans. Roy. Soc. London 151 (1861), 293--326.

2. Dickson L.E. Algebras and Their Arithmetics, University of Chicago Press, Chicago, 1923.

3. Wedderburn J.H.M. Non-commutative domains of integrity, J. Reine Angew Math. 167 (1932), 129--141.

4. Van der Waerden B.L. Modern Algebra, Springer, Berlin-New York, 1930.

5. Jacobson N. Pseudo-linear transformations, Ann. of Math. 38 (1937), 484--507.

6. Teichmüller O. Der Elementarteilsatz für nichtkommutative Ringe, Abh. Preuss. Acad. Wiss. Phys.-Math. Kl. (1937), 169–177.

7. Asano K. Neichtkommutative Hauptidealringe, Act. Sci. Ind. 696 (1938), Hermann, Paris.

8. Jacobson N. The Theory of Rings, American Mathematical Society, Mathematical Surveys 2, Providence, R.1. 1943.

9. Levy L. Robson C.J. Matrices and Pairs of Modules, Journal of Algebra, 29 (1974), 427--454.

10. Kaplansky I. Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949), 464--491.

11. Larsen M., Lewis N., Shores T. Elementary divisor rings and finitely presented modules. Trans. Amer. Math. Soc. V. 187, no.1 (1974), 231--248.

12. Gillman L., Henriksen M. Rings of continous functions in which every finitely generated ideal is principal, Trans.Amer.Math.Soc. 82 (1956), 366--391.

13. Amitsur S.A. Remarks of principal ideal rings. Osaka Math. Journ. 15 (1963), 59--69.

14. Henriksen M. Some remarks on elementary divisor rings. 11, Michigan Math. J. 3 (1955/56), 159--163.

15. Helmer O. The elementary divisor theorem for certain rings without chain conditions, Bull. Amer. Math. Soc. 49 (1943), 225--136.

16. Henriksen M. On a class of regular rings that are elementary divisor rings, Arch. Math. 24 (1973), 133--141.

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

18. Ara P. Goodearl K. O'Meara K.C., Pardo E. Diagonalization of matrices over regular rings. Linear Algebra Appl. 265 (1987), 147--163.

19. Забавський Б.В., Комарницький М.Я. Дистрибутивні області елементарних дільників, Укр. мат. журн., 42 (1990), № 7, 1000--1004.

20. Tuganbaev A.A. Semidistributive Modules and Rings, Kluwer Academic Publ. Netherland, 1998.

21. Забавський Б.В. Редукція матриць над кільцями Безу, стабільний ранг яких не перевищує 2, Укр. мат. журн., 55 (2003), № 4, 550--554.

22. Туганбаев А.А. Кольца элементарных делителей и дистрибутивные кольца, Успехи мат. наук, 46 (1991), № 6, 219--220.

23. Забавский Б.В. О некоммутативных кольцах элементарных делителей, Укр. мат. журн., 39 (1987), № 4, 440--449.

24. Zabavsky B. Diagonalizability theorems for matrices over rings with finite stable range, Algebra and Discrete Math. (2005), no.1, 134--148.

	Diagonalization of matrices
Author	B.V.Zabavsky Ivan Franko National University of Lviv
Abstract	We consider the question of diagonalizability of matrices of given size over regular rings of finite stable range as well as ``weak'' diagonal reduction of arbitrary regular ring.
Keywords	diagonalization, matrix, regular ring, stable range
DOI	doi:10.30970/ms.23.1.3-10
Reference	1. Smith H.J.S. On systems of linear indeterminate equations and congruences, Philos. Trans. Roy. Soc. London 151 (1861), 293--326. 2. Dickson L.E. Algebras and Their Arithmetics, University of Chicago Press, Chicago, 1923. 3. Wedderburn J.H.M. Non-commutative domains of integrity, J. Reine Angew Math. 167 (1932), 129--141. 4. Van der Waerden B.L. Modern Algebra, Springer, Berlin-New York, 1930. 5. Jacobson N. Pseudo-linear transformations, Ann. of Math. 38 (1937), 484--507. 6. Teichmüller O. Der Elementarteilsatz für nichtkommutative Ringe, Abh. Preuss. Acad. Wiss. Phys.-Math. Kl. (1937), 169–177. 7. Asano K. Neichtkommutative Hauptidealringe, Act. Sci. Ind. 696 (1938), Hermann, Paris. 8. Jacobson N. The Theory of Rings, American Mathematical Society, Mathematical Surveys 2, Providence, R.1. 1943. 9. Levy L. Robson C.J. Matrices and Pairs of Modules, Journal of Algebra, 29 (1974), 427--454. 10. Kaplansky I. Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949), 464--491. 11. Larsen M., Lewis N., Shores T. Elementary divisor rings and finitely presented modules. Trans. Amer. Math. Soc. V. 187, no.1 (1974), 231--248. 12. Gillman L., Henriksen M. Rings of continous functions in which every finitely generated ideal is principal, Trans.Amer.Math.Soc. 82 (1956), 366--391. 13. Amitsur S.A. Remarks of principal ideal rings. Osaka Math. Journ. 15 (1963), 59--69. 14. Henriksen M. Some remarks on elementary divisor rings. 11, Michigan Math. J. 3 (1955/56), 159--163. 15. Helmer O. The elementary divisor theorem for certain rings without chain conditions, Bull. Amer. Math. Soc. 49 (1943), 225--136. 16. Henriksen M. On a class of regular rings that are elementary divisor rings, Arch. Math. 24 (1973), 133--141. 17. Menal P., Moncasi J. On regular rings with stable range 2, J. Pure Appl. Algebra 24 (1982), 25--40. 18. Ara P. Goodearl K. O'Meara K.C., Pardo E. Diagonalization of matrices over regular rings. Linear Algebra Appl. 265 (1987), 147--163. 19. Забавський Б.В., Комарницький М.Я. Дистрибутивні області елементарних дільників, Укр. мат. журн., 42 (1990), № 7, 1000--1004. 20. Tuganbaev A.A. Semidistributive Modules and Rings, Kluwer Academic Publ. Netherland, 1998. 21. Забавський Б.В. Редукція матриць над кільцями Безу, стабільний ранг яких не перевищує 2, Укр. мат. журн., 55 (2003), № 4, 550--554. 22. Туганбаев А.А. Кольца элементарных делителей и дистрибутивные кольца, Успехи мат. наук, 46 (1991), № 6, 219--220. 23. Забавский Б.В. О некоммутативных кольцах элементарных делителей, Укр. мат. журн., 39 (1987), № 4, 440--449. 24. Zabavsky B. Diagonalizability theorems for matrices over rings with finite stable range, Algebra and Discrete Math. (2005), no.1, 134--148.
Pages	3-10
Volume	23
Issue	1
Year	2005
Journal	Matematychni Studii
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