<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this article, we formalize in Mizar [7] the definition of “torsion part” of ℤ-module and its properties. We show ℤ-module generated by the field of rational numbers as an example of torsion-free non free ℤ-modules. We also formalize the rank-nullity theorem over finite-rank free ℤ-modules (previously formalized in [1]). ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [23] and cryptographic systems with lattices [24].</p></abstract>

In this article, we formalize in Mizar [7] the definition of “torsion part” of ℤ-module and its properties. We show ℤ-module generated by the field of rational numbers as an example of torsion-free non free ℤ-modules. We also formalize the rank-nullity theorem over finite-rank free ℤ-modules (previously formalized in [1]). ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [23] and cryptographic systems with lattices [24].

Torsion Part of ℤ-module

Japan Advanced Institute of Science and Technology, Ishikawa, Japan

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

{"article-title":"Torsion Part of ℤ-module"}

In this article, we formalize in Mizar [7] the definition of “torsion part” of ℤ-module and its properties. We show ℤ-module generated by the field of rational...

Torsion Part of ℤ-module

Online veröffentlicht: 25. März 2016

Seitenbereich: 297 - 307

Eingereicht: 14. Aug. 2015

DOI: https://doi.org/10.1515/forma-2015-0024

Schlüsselwörter
torsion part of ℤ-module, torsion-free non free ℤ-module

© 2015 Yuichi Futa et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

Torsion Part of ℤ-module

Online veröffentlicht: 25. März 2016

Seitenbereich: 297 - 307

Eingereicht: 14. Aug. 2015

DOI: https://doi.org/10.1515/forma-2015-0024

Schlüsselwörtertorsion part of ℤ-module, torsion-free non free ℤ-module

© 2015 Yuichi Futa et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

Schlüsselwörter
torsion part of ℤ-module, torsion-free non free ℤ-module