Skip to main content
Log in

Grothendieck and Whitehead Groups of Formal Matrix Rings

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

For formal matrix rings, we construct some method of calculation of groups K 0 and K 1 with the use of groups K 0 and K 1 of original rings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. N. Abyzov and D. T. Tapkin, “On certain classes of rings of formal matrices,” Russ. Math., 59, No. 3 (2015).

  2. H. Bass, Algebraic K-Theory, Benjamin, New York (1968).

  3. D. Benkovič, “Lie derivations on triangular matrices,” Linear Multilinear Algebra, 55, 619–626 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Benkovič and D. Eremita, “Commuting traces and commutativity preserving maps on triangular algebras,” J. Algebra, 280, 797–824 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  5. C. Boboc, S. Dăscălescu, and L. van Wyk, “Isomorphisms between Morita context rings,” Linear Multilinear Algebra, 60, No. 5, 545–563 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Brešar, “Commuting maps: A survey,” Taiwan. J. Math., 8, 361–397 (2004).

    MathSciNet  MATH  Google Scholar 

  7. H. Chen, “Morita contexts with many units,” Commun. Algebra, 30, No. 3, 1499–1512 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  8. R. K. Dennis and S. Geller, “K i of upper triangular matrix rings,” Proc. Am. Math. Soc., 56, 73–78 (1976).

    MathSciNet  MATH  Google Scholar 

  9. E. Enochs and B. Torrecillas, “Flat covers over formal triangular matrix rings and minimal Quillen factorizations,” Forum Math., 23, 611–624 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  10. Z. Hao and K.-P. Shum, “The Grothendieck group of rings of Morita contexts,” Proc. 1996 Beijing Int. Conf. on Group Theory (1997), pp. 88–97.

  11. R. Khazal, S. Dăscălescu, and L. van Wyk, “Isomorphism of generalized triangular matrix rings and recovery of tiles,” Int. J. Math. Math. Sci., No. 9, 533–538 (2003).

  12. P. A. Krylov, “Isomorphisms of generalized matrix rings,” Algebra Logic, 47, No. 4, 258–262 (2008).

    Article  MathSciNet  Google Scholar 

  13. P. A. Krylov, “The group K 0 of a generalized matrix ring,” Algebra Logic, 52, No. 3, 250–261 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  14. P. A. Krylov, “Calculation of the group K 1 of a generalized matrix ring,” Sib. Math. J., 55, No. 4, 639–644 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  15. P. A. Krylov, A. V. Mikhalev, and A. A. Tuganbaev, Endomorphism Rings of Abelian Groups, Kluwer Academic, Dordrecht (2003).

    Book  MATH  Google Scholar 

  16. P. A. Krylov and A. A. Tuganbaev, “Modules over formal matrix rings,” J. Math. Sci., 171, No. 2, 248–295 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  17. P. A. Krylov and A. A. Tuganbaev, “Formal matrices and their determinants,” J. Math. Sci., 211, No. 3, 341–380 (2015).

    Article  MATH  Google Scholar 

  18. Y.-B. Li and F. Wei, “Semi-centralizing maps of generalized matrix algebras,” Linear Algebra Appl., 436, 1122–1153 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  19. X. Liang, F. Wei, Z. Xiao, and A. Fošner, “Centralizing traces and Lie triple isomorphisms on triangular algebras,” Linear Multilinear Algebra, 8, No. 3, 821–847 (2014).

    MATH  Google Scholar 

  20. S. K. Nauman, “Morita similar matrix rings and their Grothendieck groups,” Aligarh Bull. Math., 23, No. 1-2, 49–60 (2004).

    MathSciNet  Google Scholar 

  21. J. Rosenberg, Algebraic K-Theory and Its Applications, Springer, Berlin (1994).

    Book  MATH  Google Scholar 

  22. Z. Xiao and F. Wei, “Commuting mappings of generalized matrix algebras,” Linear Algebra Appl., 433, 2178–2197 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  23. Z. Xiao and F. Wei, “Commuting traces and Lie isomorphisms on generalized matrix algebras,” Operators Matrices, 8, 821–847 (2014).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to P. A. Krylov.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 20, No. 1, pp. 173–203, 2015.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Krylov, P.A., Tuganbaev, A.A. Grothendieck and Whitehead Groups of Formal Matrix Rings. J Math Sci 223, 606–628 (2017). https://doi.org/10.1007/s10958-017-3370-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-017-3370-z

Navigation