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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
Online
ISSN
1898-9934
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Volume 24, Issue 1 (Mar 2016)

Issues

Divisible ℤ-modules

Yuichi Futa
  • Corresponding author
  • Japan Advanced Institute of Science and Technology Ishikawa, Japan
/ Yasunari Shidama
  • Shinshu University Nagano, Japan
Published Online: 2016-08-31 | DOI: https://doi.org/10.1515/forma-2016-0004

Summary

In this article, we formalize the definition of divisible ℤ-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible ℤ-modules are not finitely-generated.We introduce a divisible ℤ-module, equivalent to a vector space of a torsion-free ℤ-module with a coefficient ring ℚ. ℤ-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [15], cryptographic systems with lattices [16] and coding theory [8].

MSC: 15A03; 16D20; 13C13; 03B35

Keywords: divisible vector; divisible ℤ-module

MML : identifier: ZMODUL08; version: 8.1.04 5.36.1267

References

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About the article

Received: 2015-12-30

Published Online: 2016-08-31

Published in Print: 2016-03-01


Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2016-0004.

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© by Yuichi Futa. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. BY-SA 3.0

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