Free Z - module 1 Yuichi Futa Shinshu University Nagano , Japan Hiroyuki Okazaki Shinshu University Nagano , Japan Yasunari Shidama Shinshu University Nagano , Japan

In this article we formalize a free Z-module and its rank. We formally prove that for a free finite rank Z-module V , the number of elements in its basis, that is a rank of the Z-module, is constant regardless of the selection of its basis. Z-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [15]. Some theorems in this article are described by translating theorems in [21] and [8] into theorems of Z-module.