Have a personal or library account? Click to login
Torsion Part of ℤ-module Cover

Torsion Part of ℤ-module

Formalized Mathematics
Volume 23 (2015): Issue 4 (December 2015)
By: Yuichi Futa,  Hiroyuki Okazaki and  Yasunari Shidama  
Open Access
|Mar 2016

References

  1. [1] Jesse Alama. The rank+nullity theorem. Formalized Mathematics, 15(3):137–142, 2007. doi:10.2478/v10037-007-0015-6.10.2478/v10037-007-0015-6
    Search in Google Scholar
  2. [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.
    Search in Google Scholar
  3. [3] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.
    Search in Google Scholar
  4. [4] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.
    Search in Google Scholar
  5. [5] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.
    Search in Google Scholar
  6. [6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.
    Search in Google Scholar
  7. [7] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17.10.1007/978-3-319-20615-8
    Search in Google Scholar
  8. [8] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.
    Search in Google Scholar
  9. [9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.
    Search in Google Scholar
  10. [10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.
    Search in Google Scholar
  11. [11] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.
    Search in Google Scholar
  12. [12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.
    Search in Google Scholar
  13. [13] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.
    Search in Google Scholar
  14. [14] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. ℤ-modules. Formalized Mathematics, 20(1):47–59, 2012. doi:10.2478/v10037-012-0007-z.10.2478/v10037-012-0007-z
    Search in Google Scholar
  15. [15] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Quotient module of ℤ-module. Formalized Mathematics, 20(3):205–214, 2012. doi:10.2478/v10037-012-0024-y.10.2478/v10037-012-0024-y
    Search in Google Scholar
  16. [16] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Free ℤ-module. Formalized Mathematics, 20(4):275–280, 2012. doi:10.2478/v10037-012-0033-x.10.2478/v10037-012-0033-x
    Search in Google Scholar
  17. [17] Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, and Yasunari Shidama. Gaussian integers. Formalized Mathematics, 21(2):115–125, 2013. doi:10.2478/forma-2013-0013.10.2478/forma-2013-0013
    Search in Google Scholar
  18. [18] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Submodule of free ℤ-module. Formalized Mathematics, 21(4):273–282, 2013. doi:10.2478/forma-2013-0029.10.2478/forma-2013-0029
    Search in Google Scholar
  19. [19] Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, and Yasunari Shidama. Torsion ℤ-module and torsion-free ℤ-module. Formalized Mathematics, 22(4):277–289, 2014. doi:10.2478/forma-2014-0028.10.2478/forma-2014-0028
    Search in Google Scholar
  20. [20] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841–845, 1990.
    Search in Google Scholar
  21. [21] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.
    Search in Google Scholar
  22. [22] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829–832, 1990.
    Search in Google Scholar
  23. [23] A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4), 1982.10.1007/BF01457454
    Search in Google Scholar
  24. [24] Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: a cryptographic perspective. The International Series in Engineering and Computer Science, 2002.10.1007/978-1-4615-0897-7_8
    Search in Google Scholar
  25. [25] Michał Muzalewski. Rings and modules – part II. Formalized Mathematics, 2(4):579–585, 1991.
    Search in Google Scholar
  26. [26] Kazuhisa Nakasho, Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Rank of submodule, linear transformations and linearly independent subsets of ℤ-module. Formalized Mathematics, 22(3):189–198, 2014. doi:10.2478/forma-2014-0021.10.2478/forma-2014-0021
    Search in Google Scholar
  27. [27] Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559–564, 2001.
    Search in Google Scholar
  28. [28] Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29–34, 1999.
    Search in Google Scholar
  29. [29] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115–122, 1990.
    Search in Google Scholar
  30. [30] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.
    Search in Google Scholar
  31. [31] Wojciech A. Trybulec. Operations on subspaces in real linear space. Formalized Mathematics, 1(2):395–399, 1990.
    Search in Google Scholar
  32. [32] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.
    Search in Google Scholar
  33. [33] Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. Formalized Mathematics, 1(5):865–870, 1990.
    Search in Google Scholar
  34. [34] Wojciech A. Trybulec. Operations on subspaces in vector space. Formalized Mathematics, 1(5):871–876, 1990.
    Search in Google Scholar
  35. [35] Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1 (5):877–882, 1990.
    Search in Google Scholar
  36. [36] Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883–885, 1990.
    Search in Google Scholar
  37. [37] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.
    Search in Google Scholar
  38. [38] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.
    Search in Google Scholar
  39. [39] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.
    Search in Google Scholar
DOI: https://doi.org/10.1515/forma-2015-0024 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630
Journal RSS Feed
Language: English
Page range: 297 - 307
Submitted on: Aug 14, 2015
|
Published on: Mar 25, 2016
Published by: University of Białystok
In partnership with: Paradigm Publishing Services
Publication frequency: 1 issue per year
Keywords:
15A03,
13C12,
03B35
Related subjects:
Mathematics,
General mathematics,
Computer sciences,
Computer sciences, other

© 2016 Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, published by University of Białystok
This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

Previous articleVolume 23 (2015): Issue 4 (December 2015)Next article