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].
@article{bwmeta1.element.doi-10_1515_forma-2016-0004,
author = {Yuichi Futa and Yasunari Shidama},
title = {Divisible $\mathbb{Z}$-modules},
journal = {Formalized Mathematics},
volume = {24},
year = {2016},
pages = {37-47},
zbl = {1343.13014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_forma-2016-0004}
}
Yuichi Futa; Yasunari Shidama. Divisible ℤ-modules. Formalized Mathematics, Tome 24 (2016) pp. 37-47. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_forma-2016-0004/
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