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].
@article{bwmeta1.element.doi-10_1515_forma-2015-0024,
author = {Yuichi Futa and Hiroyuki Okazaki and Yasunari Shidama},
title = {Torsion Part of $\mathbb{Z}$-module},
journal = {Formalized Mathematics},
volume = {23},
year = {2015},
pages = {297-307},
zbl = {1334.13008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_forma-2015-0024}
}
Yuichi Futa; Hiroyuki Okazaki; Yasunari Shidama. Torsion Part of ℤ-module. Formalized Mathematics, Tome 23 (2015) pp. 297-307. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_forma-2015-0024/
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