Rings with the Kazimirsky condition and rings with projective
socle

B. V. Zabavsky; O. M. Romaniv

doi:10.15330/ms.51.1.124-129

We construct the theory of diagonalizability for matrices over Bezout rings of stable range 1 with the Kazimirsky condition. It is shown that a ring of stable range 1 with the right (left) Kazimirsky condition is an elementary divisor ring if and only if it is a duo ring.We describe the conditions under which a proper finite homomorphic image of a commutative Bezout domain is a ring with projective socle.

1. G. Baccella, On &-semisimple rings. A study of the socle of a ring, Comm. Alg., 8 (1980), №10, 889–909.

2. V. Camillo, W.K. Nicholson, Z. Wang, Left quasi-morphic rings, J. Alg. Appl., 7 (2008), №6, 725–733.

3. H. Chen, Rings related to stable range conditions Series in Algebra 11, World Scientific, Hackensack, NJ, 2011.

4. R.R. Colby, Rings which have flat injective modules, J. Alg., 35 (1975) 239–252.

5. N.I. Dubrovin, On noncommutative rings with elementary divisors, Reports of Institutes of Higher Education Math., 11 (1986), 14–20.

6. G.A. Garkusha, FP-injective and weakly quasi-Frobenius rings, J. Math. Scien., 112 (2002), №3, 4303– 4312.

7. R. Gordon, Rings in which minimal left ideals are projective, Pacific J. Math., 31 (1969), №3, 679–692.

8. M. Henriksen, On a class of regular rings that are elementary divisor rings, Archiv Math., 24 (1973), №1, 133–141.

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

10. D. Khurana, G. Marks, A.K. Srivastava, On unit-central rings, Springer, Advances in Ring Theory, Trends in Mathematics, Birkhauser Verlag-Based/Switzerland, 2010, 205–212.

11. G. Marks, Duo rings and Ore extensions, J. Alg., 280 (2004), №2, 463–471.

12. W.K. Nicholson, J.F. Watters, Rings with projective socle, Proc. Amer. Math. Soc., 102 (1988), №3, 443–450.

13. M. Satyanarayana, Rings with primary ideals as maximal ideals, Math. Scand., 20 (1967), 52–54.

14. A.A. Tuganbaev, Rings of elementary divisors and distributive rings, Russ. Math. Surv., 46 (1991), №6, 219–220.

15. B.V. Zabavsky, M.Ya. Komarnytsky, Distributive elementary divisor domains, Ukr. Math. J., 42 (1990), №7, 890–892.

16. B. V. Zabavsky, Diagonal reduction of matrices over rings, Mathematical Studies, Monograph Series, V.XVI, VNTL Publishers, Lviv, 2012.

17. B. Zabavsky, O. Pihura, Bezout morphic rings, Visnyk Lviv Univ., 79 (2014), 163–168.

	Rings with the Kazimirsky condition and rings with projective socle
Author	B. V. Zabavsky¹, O. M. Romaniv² zabavskii@gmail.com¹, oleh.romaniv@lnu.edu.ua² Department of Mechanics and Mathematics Ivan Franko National University, Lviv, Ukraine
Abstract	We construct the theory of diagonalizability for matrices over Bezout rings of stable range 1 with the Kazimirsky condition. It is shown that a ring of stable range 1 with the right (left) Kazimirsky condition is an elementary divisor ring if and only if it is a duo ring.We describe the conditions under which a proper finite homomorphic image of a commutative Bezout domain is a ring with projective socle.
Keywords	Bezout ring; Kazimirsky condition; elementary divisor ring; stable range; unit stable range; projective socle; maximal ideal; Kasch ring; PS-ring; PS-module; unit-central ring; duo ring
DOI	doi:10.15330/ms.51.2.124-129
Reference	1. G. Baccella, On &-semisimple rings. A study of the socle of a ring, Comm. Alg., 8 (1980), №10, 889–909. 2. V. Camillo, W.K. Nicholson, Z. Wang, Left quasi-morphic rings, J. Alg. Appl., 7 (2008), №6, 725–733. 3. H. Chen, Rings related to stable range conditions Series in Algebra 11, World Scientific, Hackensack, NJ, 2011. 4. R.R. Colby, Rings which have flat injective modules, J. Alg., 35 (1975) 239–252. 5. N.I. Dubrovin, On noncommutative rings with elementary divisors, Reports of Institutes of Higher Education Math., 11 (1986), 14–20. 6. G.A. Garkusha, FP-injective and weakly quasi-Frobenius rings, J. Math. Scien., 112 (2002), №3, 4303– 4312. 7. R. Gordon, Rings in which minimal left ideals are projective, Pacific J. Math., 31 (1969), №3, 679–692. 8. M. Henriksen, On a class of regular rings that are elementary divisor rings, Archiv Math., 24 (1973), №1, 133–141. 9. I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc., 66 (1949) 464–491. 10. D. Khurana, G. Marks, A.K. Srivastava, On unit-central rings, Springer, Advances in Ring Theory, Trends in Mathematics, Birkhauser Verlag-Based/Switzerland, 2010, 205–212. 11. G. Marks, Duo rings and Ore extensions, J. Alg., 280 (2004), №2, 463–471. 12. W.K. Nicholson, J.F. Watters, Rings with projective socle, Proc. Amer. Math. Soc., 102 (1988), №3, 443–450. 13. M. Satyanarayana, Rings with primary ideals as maximal ideals, Math. Scand., 20 (1967), 52–54. 14. A.A. Tuganbaev, Rings of elementary divisors and distributive rings, Russ. Math. Surv., 46 (1991), №6, 219–220. 15. B.V. Zabavsky, M.Ya. Komarnytsky, Distributive elementary divisor domains, Ukr. Math. J., 42 (1990), №7, 890–892. 16. B. V. Zabavsky, Diagonal reduction of matrices over rings, Mathematical Studies, Monograph Series, V.XVI, VNTL Publishers, Lviv, 2012. 17. B. Zabavsky, O. Pihura, Bezout morphic rings, Visnyk Lviv Univ., 79 (2014), 163–168.
Pages	124-129
Volume	51
Issue	2
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Rings with the Kazimirsky condition and rings with projective socle