Torsion Z-module and Torsion-free Z-module

Yuichi Futa; Hiroyuki Okazaki; Kazuhisa Nakasho; Yasunari Shidama

Formalized Mathematics (2014)

  • Volume: 22, Issue: 4, page 277-289
  • ISSN: 1426-2630

Abstract

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In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].

How to cite

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Yuichi Futa, et al. "Torsion Z-module and Torsion-free Z-module." Formalized Mathematics 22.4 (2014): 277-289. <http://eudml.org/doc/271001>.

@article{YuichiFuta2014,
abstract = {In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].},
author = {Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, Yasunari Shidama},
journal = {Formalized Mathematics},
keywords = {free Z-module; rank of Z-module; homomorphism of Z-module; linearly independent; linear combination; free -module; rank of -module; homomorphism of -module},
language = {eng},
number = {4},
pages = {277-289},
title = {Torsion Z-module and Torsion-free Z-module},
url = {http://eudml.org/doc/271001},
volume = {22},
year = {2014},
}

TY - JOUR
AU - Yuichi Futa
AU - Hiroyuki Okazaki
AU - Kazuhisa Nakasho
AU - Yasunari Shidama
TI - Torsion Z-module and Torsion-free Z-module
JO - Formalized Mathematics
PY - 2014
VL - 22
IS - 4
SP - 277
EP - 289
AB - In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].
LA - eng
KW - free Z-module; rank of Z-module; homomorphism of Z-module; linearly independent; linear combination; free -module; rank of -module; homomorphism of -module
UR - http://eudml.org/doc/271001
ER -

References

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