Affiliation:
1. Japan Advanced Institute of Science and Technology Ishikawa, Japan
2. Shinshu University Nagano, Japan
Abstract
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].
Subject
Applied Mathematics,Computational Mathematics
Cited by
5 articles.
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1. General properties of hollow modules;THE 2ND UNIVERSITAS LAMPUNG INTERNATIONAL CONFERENCE ON SCIENCE, TECHNOLOGY, AND ENVIRONMENT (ULICoSTE) 2021;2022
2. Distributive Rings and Some Domains;Journal of Physics: Conference Series;2021-07-01
3. A New Results of Injective Module with Divisible Property;Journal of Physics: Conference Series;2021-03-01
4. Dual Lattice of ℤ-module Lattice;Formalized Mathematics;2017-07-01
5. Embedded Lattice and Properties of Gram Matrix;Formalized Mathematics;2017-03-28