Purity and direct summands.

Oneto R., Ángel V.

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A short note on the non-negativity of partial Euler characteristics.

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Z-modules

Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2012)

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In this article, we formalize Z-module, that is a module over integer ring. Z-module is necassary for lattice problems, LLL (Lenstra-Lenstra-Lovász) base reduction algorithm and cryptographic systems with lattices [11].

Submodule of free Z-module

Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2013)

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In this article, we formalize a free Z-module and its property. In particular, we formalize the vector space of rational field corresponding to a free Z-module and prove formally that submodules of a free Z-module are free. Z-module is necassary for lattice problems - LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattice [20]. Some theorems in this article are described by translating theorems in [11] into theorems of Z-module, however their...