Questions related to the K-theoretical aspect of Bezout rings with various stable
range conditions

B. V. Zabavsky

We provide a list of open problems that are connected to the commutative and noncommutative ring theory, K-theory and homological algebra. Some problems are solved now completely, some partially, and most of them, remaining still open, are supplemented with ideas and hints.

1. Ara P., Goodearl K.R., O’Meara K.C., Pardo E. Separative cancellation for projective modules over exchange rings// Israel J. Math. – 1998. – V.105, №1. – P. 105–137.

2. Bass H. K-theory and stable algebra// Publ. Math. – 1964. – V.22. – P. 5–60.

3. Colby R.R. Rings which have flat injective modules// J. Algebra. – V.35. – 1975. – P. 239–252.

4. Cohn P., Free rings and their relations. – Moscow: Mir., 1976. (in Russian)

5. Contessa M. On pm-rings// Comm. Algebra. – 1982. – V.10, №1. – P. 93–108.

6. Contessa M. On certain classes of PM-rings// Communications in Algebra. – 1984. – V.12. – P. 1447– 1469.

7. De Marco G., Orsati A. Commutative rings in which every prime ideal is contained in a unique maximal ideal// Proc. Amer. Math. Soc. – 1971. – V.30, №3. – P. 459–466.

8. Garkusha G.A. FP-injective and weakly quasi-frobenius rings// Zapiski nauch. sem. POMI. – 1999. – V.265. – P. 110–129. (in Russian)

9. Gatalevych A. On adequate and generalized adequate duo-rings and elementary divisor duo-rings// Mat. Stud. – 1998. – V.9. – P. 115–119.

10. Gillman L., Henriksen M. Rings of continuous functions in which every finitely generated ideal is principal// Trans. Amer. Math. Soc. – 1956. – V.82, №2. – P. 366–391.

11. Glaz S. Commutative Coherent Rings. – Springer-Verlag, 1989. – 347p.

12. Goodearl K.R. Von Neumann Regular Rings. – London: Pitman, 1979. – 369p.

13. Henriksen M. Some remarks on elementary divisor rings II// Michigan Math. J. – 1955. – V.3, №2. – P. 159–163.

14. Jondrup S. Rings in which pure ideals are generated by idempotents// Math. Scand. – 1972. – V.30 – P. 177–185.

15. Kaplansky I. Elementary divisors and modules// Trans. Amer. Math. Soc. – 1949. – V.66. – P. 464–491.

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

17. Lam T.Y., Dugas A.S. Quasi-duo rings and stable range descent// J. Pure Appl. Alg. – 2005. – V.195. – P. 243–259.

18. McGovern W.W. Neat rings// J. Pure Appl. Algebra. – V.205, №2. – 2006. – P. 243–265.

19. Nicholson W.K. Lifting idempotents and exchange rings// Trans. Amer. Alath. Soc. – 1977. – V.229. – P. 269–278.

20. Nicholson W.K. Rings whose elements are quasi-regular or regular// Aequationes Mathematicae. – 1973. – V.9 – P. 64–70.

21. Nicholson W.K., Sanchez Campos E. Rings with the dual of the isomorphism theorem// J. Algebra. – 2004. – V.271. – P. 391–406.

22. Nicholson W.K., Yousif M.F., Quasi-Frobenius rings. – Cambridge University Press, 2003.

23. Roitman J., An introduction to homological algebra. – Academic Press, 1979.

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

25. Stenstrom B. Coherent rings and FP-injective modules// J. London Math. Soc. – V.2. – 1970. – P. 323– 329.

26. Tuganbaev A.A. Elementary divisor rings and distributive rings// Uspekhi. Mat. Nauk. – 1991. – V.46, №6. – P. 219–220. (in Russian)

27. Tuganbaev A.A., Rings theory. Arithmetical modules and rings. – Miscow: MTsNMO, 2009. – 472p. (in Russian)

28. Vamos P. 2-good rings// Quart. J. Math. – 2005. – V.56. – P. 417–430.

29. Vasiunyk I.S., Zabavsky B.V. Rings of almost unit stable range one// Ukr. Mat. Zh. – 2011. – V.63, №6. – P. 840–843. (in Ukrainian)

30. Vasserstein L.N. The stable rank of rings and dimensionality of topological spaces// Functional Anal. Appl. – 1971. – V.5. – P. 102–110.

31. Yu H.-P. On quasi-duo rings// Glasgow Math. J. – 1995. – V.37. – P. 21–31.

32. Zabavsky B.V. Diagonal reduction of matrices over rings. Mathematical Studies: Monograph Series. V.XVI. – Lviv: VNTL Publishers, 2012. – 251 p.

33. Zabavsky B.V. Fractionally regular Bezout rings// Mat. Stud. – 2009. – V.32, №1. – P. 76–80.

34. Zabavsky B.V. Diagonal reduction of matrices over finite stable range rings// Mat. Stud. – 2014. – V.41, №1. – P. 101–108.

35. Zabavsky B.V., Bilyavs’ka S.I. Every zero adequate ring is an exchange ring// Fundam. Prikl. Mat. – 2012. – V.17, №3 – P. 61–66. (in Russian)

36. Zabavsky B.V., Bilavska S.I.Weak global dimension of finite homomorphic images of commutative Bezout domain// Prykl. Probl. Mat. and Mech. – 2012. – V.10. – P. 71–73. (in Ukrainian)

37. Zabavsky B.V., Komarnytskii N.Ya. Distributive domains with elementary divisors// Ukr. Mat. Zh. – 1990. – V.42, №7. – P. 1000–1004. (in Russian)

	Questions related to the K-theoretical aspect of Bezout rings with various stable range conditions
Author	B. V. Zabavsky zabavskii@gmail.com Ivan Franko National University of Lviv
Abstract	We provide a list of open problems that are connected to the commutative and noncommutative ring theory, K-theory and homological algebra. Some problems are solved now completely, some partially, and most of them, remaining still open, are supplemented with ideas and hints.
Keywords	Bezout ring; K-theory; elementary divisor ring; Hermite ring; stable range; exchange ring; adequate ring; morphic ring
Reference	1. Ara P., Goodearl K.R., O’Meara K.C., Pardo E. Separative cancellation for projective modules over exchange rings// Israel J. Math. – 1998. – V.105, №1. – P. 105–137. 2. Bass H. K-theory and stable algebra// Publ. Math. – 1964. – V.22. – P. 5–60. 3. Colby R.R. Rings which have flat injective modules// J. Algebra. – V.35. – 1975. – P. 239–252. 4. Cohn P., Free rings and their relations. – Moscow: Mir., 1976. (in Russian) 5. Contessa M. On pm-rings// Comm. Algebra. – 1982. – V.10, №1. – P. 93–108. 6. Contessa M. On certain classes of PM-rings// Communications in Algebra. – 1984. – V.12. – P. 1447– 1469. 7. De Marco G., Orsati A. Commutative rings in which every prime ideal is contained in a unique maximal ideal// Proc. Amer. Math. Soc. – 1971. – V.30, №3. – P. 459–466. 8. Garkusha G.A. FP-injective and weakly quasi-frobenius rings// Zapiski nauch. sem. POMI. – 1999. – V.265. – P. 110–129. (in Russian) 9. Gatalevych A. On adequate and generalized adequate duo-rings and elementary divisor duo-rings// Mat. Stud. – 1998. – V.9. – P. 115–119. 10. Gillman L., Henriksen M. Rings of continuous functions in which every finitely generated ideal is principal// Trans. Amer. Math. Soc. – 1956. – V.82, №2. – P. 366–391. 11. Glaz S. Commutative Coherent Rings. – Springer-Verlag, 1989. – 347p. 12. Goodearl K.R. Von Neumann Regular Rings. – London: Pitman, 1979. – 369p. 13. Henriksen M. Some remarks on elementary divisor rings II// Michigan Math. J. – 1955. – V.3, №2. – P. 159–163. 14. Jondrup S. Rings in which pure ideals are generated by idempotents// Math. Scand. – 1972. – V.30 – P. 177–185. 15. Kaplansky I. Elementary divisors and modules// Trans. Amer. Math. Soc. – 1949. – V.66. – P. 464–491. 16. Khurana D., Marks G., Srivastava A.K. On unit-central rings// Springer: Advances in Ring Theory, Trends in Mathematics, Birkhauser Verlag Basel/Switzerland, 2010. - P. 205–212. 17. Lam T.Y., Dugas A.S. Quasi-duo rings and stable range descent// J. Pure Appl. Alg. – 2005. – V.195. – P. 243–259. 18. McGovern W.W. Neat rings// J. Pure Appl. Algebra. – V.205, №2. – 2006. – P. 243–265. 19. Nicholson W.K. Lifting idempotents and exchange rings// Trans. Amer. Alath. Soc. – 1977. – V.229. – P. 269–278. 20. Nicholson W.K. Rings whose elements are quasi-regular or regular// Aequationes Mathematicae. – 1973. – V.9 – P. 64–70. 21. Nicholson W.K., Sanchez Campos E. Rings with the dual of the isomorphism theorem// J. Algebra. – 2004. – V.271. – P. 391–406. 22. Nicholson W.K., Yousif M.F., Quasi-Frobenius rings. – Cambridge University Press, 2003. 23. Roitman J., An introduction to homological algebra. – Academic Press, 1979. 24. Shores T. Modules over semihereditary Bezout rings// Proc. Amer. Math. Soc. – V.46, №2. – 1974. – P. 211–213. 25. Stenstrom B. Coherent rings and FP-injective modules// J. London Math. Soc. – V.2. – 1970. – P. 323– 329. 26. Tuganbaev A.A. Elementary divisor rings and distributive rings// Uspekhi. Mat. Nauk. – 1991. – V.46, №6. – P. 219–220. (in Russian) 27. Tuganbaev A.A., Rings theory. Arithmetical modules and rings. – Miscow: MTsNMO, 2009. – 472p. (in Russian) 28. Vamos P. 2-good rings// Quart. J. Math. – 2005. – V.56. – P. 417–430. 29. Vasiunyk I.S., Zabavsky B.V. Rings of almost unit stable range one// Ukr. Mat. Zh. – 2011. – V.63, №6. – P. 840–843. (in Ukrainian) 30. Vasserstein L.N. The stable rank of rings and dimensionality of topological spaces// Functional Anal. Appl. – 1971. – V.5. – P. 102–110. 31. Yu H.-P. On quasi-duo rings// Glasgow Math. J. – 1995. – V.37. – P. 21–31. 32. Zabavsky B.V. Diagonal reduction of matrices over rings. Mathematical Studies: Monograph Series. V.XVI. – Lviv: VNTL Publishers, 2012. – 251 p. 33. Zabavsky B.V. Fractionally regular Bezout rings// Mat. Stud. – 2009. – V.32, №1. – P. 76–80. 34. Zabavsky B.V. Diagonal reduction of matrices over finite stable range rings// Mat. Stud. – 2014. – V.41, №1. – P. 101–108. 35. Zabavsky B.V., Bilyavs’ka S.I. Every zero adequate ring is an exchange ring// Fundam. Prikl. Mat. – 2012. – V.17, №3 – P. 61–66. (in Russian) 36. Zabavsky B.V., Bilavska S.I.Weak global dimension of finite homomorphic images of commutative Bezout domain// Prykl. Probl. Mat. and Mech. – 2012. – V.10. – P. 71–73. (in Ukrainian) 37. Zabavsky B.V., Komarnytskii N.Ya. Distributive domains with elementary divisors// Ukr. Mat. Zh. – 1990. – V.42, №7. – P. 1000–1004. (in Russian)
Pages	89-103
Volume	42
Issue	1
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Questions related to the K-theoretical aspect of Bezout rings with various stable range conditions