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Submodule of free Z-module Cover

Submodule of free Z-module

Formalized Mathematics
Volume 21 (2013): Issue 4 (December 2013)
By: Yuichi Futa,  Hiroyuki Okazaki and  Yasunari Shidama  
Open Access
|Dec 2013

References

  1. [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
    Search in Google Scholar
  2. [2] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.
    Search in Google Scholar
  3. [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
    Search in Google Scholar
  4. [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
    Search in Google Scholar
  5. [5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
    Search in Google Scholar
  6. [6] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
    Search in Google Scholar
  7. [7] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
    Search in Google Scholar
  8. [8] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
    Search in Google Scholar
  9. [9] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
    Search in Google Scholar
  10. [10] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
    Search in Google Scholar
  11. [11] Jing-Chao Chen. The Steinitz theorem and the dimension of a real linear space. Formalized Mathematics, 6(3):411-415, 1997.
    Search in Google Scholar
  12. [12] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
    Search in Google Scholar
  13. [13] 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
  14. [14] 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
  15. [15] 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
  16. [16] 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
  17. [17] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.
    Search in Google Scholar
  18. [18] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
    Search in Google Scholar
  19. [19] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.
    Search in Google Scholar
  20. [20] Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: a cryptographic perspective. 2002.10.1007/978-1-4615-0897-7
    Search in Google Scholar
  21. [21] Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339-345, 1996.
    Search in Google Scholar
  22. [22] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.
    Search in Google Scholar
  23. [23] Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29-34, 1999.
    Search in Google Scholar
  24. [24] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.
    Search in Google Scholar
  25. [25] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
    Search in Google Scholar
  26. [26] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
    Search in Google Scholar
  27. [27] Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1(5):877-882, 1990.
    Search in Google Scholar
  28. [28] Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883-885, 1990.
    Search in Google Scholar
  29. [29] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
    Search in Google Scholar
  30. [30] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
    Search in Google Scholar
  31. [31] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
    Search in Google Scholar
  32. [32] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990.
    Search in Google Scholar
  33. [33] Mariusz Zynel. The Steinitz theorem and the dimension of a vector space. Formalized Mathematics, 5(3):423-428, 1996.
    Search in Google Scholar
DOI: https://doi.org/10.2478/forma-2013-0029 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630
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Language: English
Page range: 273 - 282
Submitted on: Dec 31, 2013
|
Published on: Dec 27, 2013
Published by: University of Białystok
In partnership with: Paradigm Publishing Services
Publication frequency: 1 issue per year
Keywords:
free Z-module,
submodule of free Z-module
Related subjects:
Mathematics,
General mathematics,
Computer sciences,
Computer sciences, other

© 2013 Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, published by University of Białystok
This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

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