Skip to main content
SpringerLink
Log in

Addition Operations on the Multiplicative Monoid of Integers

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

The paper studies modules over a certain generalized ring. This ring is the noncommutative tensor square of the ring of integers. The modules in question are related to some interesting arithmetic problems. In particular, they are related to the solved Gauss class-number problem for imaginary quadratic fields.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. Durov, “New approach to Arakelov geometry,” arXiv: 0704.2030, 1 [math AG] 16 Apr 2007.

  2. A. Suslin and V. Voevodsky, “Bloch–Kato conjecture and motivic cohomology with finite coefficients,” in: The Arithmetic and Geometry of Algebraic Cycles, Kluwer Acad. Publ., Dordrecht (2000), pp. 117–189.

    Chapter  Google Scholar 

  3. N. Bourbaki, Commutative Algebra [Russian translation], Mir, Moscow (1971).

    Google Scholar 

  4. J.–P. Serre, “Local algebra and the theory of multiplicities,” Mathematics, 7, No. 5, 3–94 (1963).

  5. H. M. Stark, “On the “gap” in the theorem of Heegner,” J. Number Theory, 1, 16–27 (1969).

    Article  MathSciNet  Google Scholar 

  6. J.–P. Serre, “Complex multiplication,” in: Algebraic Number Theory, J. Cassels and A. Frohlich, eds. [Russian transslation], Mir, Moscow (1969).

  7. M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra [Russian translation], Mir, Moscow (1972).

    Google Scholar 

  8. M. Hochster, “Nonuniqueness of coefficient rings in a polynomial ring,” Proc. Amer. Math. Soc., 34, No. 1, 81–82 (1972).

    Article  MathSciNet  Google Scholar 

  9. R. Hartshorne, Algebraic Geometry [Russian translation], Mir, Moscow (1981).

    MATH  Google Scholar 

  10. J.–P. Serre, Course of Arithmetics [Russian translation], Mir, Moscow (1972).

  11. J. Tate, “The arithmetic of elliptic curves,” Invent. Math., 23, 179–206 (1974).

    Article  MathSciNet  Google Scholar 

  12. D. Goldfeld, “Gauss class number problem for imaginary quadratic fields,” Bull. Amer. Math. Soc., 13, 23–37 (1985).

    Article  MathSciNet  Google Scholar 

  13. E. W. Howe, K. E. Lauter, and J. Top, Pointless curves of genus three and four, arXiv:math/0403178v1 [math.NT] 10 Mar (2004).

  14. A. Mattuck and J. Tate, “On the inequality of Castelnuovo–Severi,” Hamb. Abh., 22, 295–299 (1958).

    Article  MathSciNet  Google Scholar 

  15. M. Deuring, “Imaginary quadratische Zahlkorper mit der Klassenzahl 1,” Math. Z., 37, 405–415 (1933).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St.Petersburg Department of V. A. Steklov Mathematical Institute of the Russian Academy of Sciences, St.Petersburg, Russia

    A. L. Smirnov

Authors
  1. A. L. Smirnov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. L. Smirnov.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 502, 2021, pp. 139–151.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Smirnov, A.L. Addition Operations on the Multiplicative Monoid of Integers. J Math Sci 264, 193–201 (2022). https://doi.org/10.1007/s10958-022-05988-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-022-05988-5

Search

Navigation