Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-10T13:47:17.075Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE WEAK GROTHENDIECK GROUP OF A BEZOUT RING

Part of: Commutative algebra: Homological methods Arithmetic rings and other special rings Grothendieck groups and $K_0$

Published online by Cambridge University Press:  21 July 2015

O. S. SOROKIN
O. S. SOROKIN*
Affiliation:
Department of Algebra and Logic, Faculty of Mechanics and Mathematics, Ivan Franko National University of L'viv, 1 Universytetska str., 79000, Lviv, Ukraine e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The K-theoretical aspect of the commutative Bezout rings is established using the arithmetical properties of the Bezout rings in order to obtain a ring of all Smith normal forms of matrices over the Bezout ring. The internal structure and basic properties of such rings are discussed as well as their presentations by the Witt vectors. In a case of a commutative von Neumann regular rings the famous Grothendieck group K0(R) obtains the alternative description.

MSC classification

Primary: 19A49: $K_0$ of other rings
Secondary: 13D15: Grothendieck groups, $K$-theory 13F35: Witt vectors and related rings
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 3 , September 2016 , pp. 617 - 635
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

REFERENCES

1. Bass, H., K-theory and stable algebra, Publ. Math. 22 (1964), 5–60.Google Scholar
2. Bourbaki, N., Algèbre, Ch. X (Masson, Paris, 1980).Google Scholar
3. Bourbaki, N., Elements de mathematique. Fase. XXVII. Algebre commutative. Chap. 1: Modules plats (Hermann, Paris, 1961).Google Scholar
4. Guralnick, R. M., Matrix equivalence and isomorphism of modules, Linear Algebra Appl. 43 (1982), 125–136.CrossRefGoogle Scholar
5. Goodearl, K. R., Von Neumann regular rings (Pitman, London, 1979), 369.Google Scholar
6. Goodearl, K. R., Torsion in K0 of unit-regular rings, Proc. Edinburgh Math. Soc. 38 (1995), 331–341.CrossRefGoogle Scholar
7. Henriksen, M., On a class of regular rings that are elementary divisor rings, Arch. Math. (Basel) 24 (1973), 133–141.Google Scholar
8. Kaplansky, I., Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949), 464–491.Google Scholar
9. Lambek, J., Lectures on rings and modules (Blaisdell Publishing, 1966).Google Scholar
10. Larsen, M., Levis, W. and Shores, T., Elementary divisor rings and finitely presented modules, Trans. Amer. Math. Soc. 187 (1974), 231–248.Google Scholar
11. Milnor, J. W., Introduction to algebraic K-theory, Ann. Math. Stud. 72 (1971), 184.Google Scholar
12. Nicholson, W. K. and Sanchez, E., Campos Rings with the dual of the isomorphism theorem, J. Algebra 271 (2004), 391–406.Google Scholar
13. Shores, T., Modules over semihereditary Bezout rings, Proc. Amer. Math. Soc. 46 (2) (1974), 211–213.CrossRefGoogle Scholar
14. Shores, T., Decomposition of modules and matrices, Bull. Amer. Math. Soc. 79 (6) (1973), 1277–1280.Google Scholar
15. Vasserstein, L. N., The stable rank of rings and dimensionality of topological spaces, Functional Anal. Appl. 5 (1971), 102–110.CrossRefGoogle Scholar
16. Warfield, R. B., Exchange rings and decompositions of modules, Math. Ann. 199 (1972), 31–36.Google Scholar
17. Weibel, C. A., The K-book: An introduction to algebraic K-theory, Grad. Stud. Math. 145 (2013), 618.CrossRefGoogle Scholar
18. Zabavsky, B. V., Diagonal reduction of matrices over rings, Mathematical Studies, Monograph Series, vol. XVI (VNTL Publishers, 2012), 251.Google Scholar
19. Zabavsky, B. V., Diagonal reduction of matrices over finite stable range rings, Math. Stud. 41 (1) (2014), 101–108.Google Scholar