Torsion-free abelian groups and completely decomposable -adic modules
G. d'Este (1980)
Rendiconti del Seminario Matematico della Università di Padova
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G. d'Este (1980)
Rendiconti del Seminario Matematico della Università di Padova
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Ulrich F. Albrecht (2001)
Czechoslovak Mathematical Journal
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Anthony Giovannitti, H. Pat Goeters, Claudia Metelli (1989)
Commentationes Mathematicae Universitatis Carolinae
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Lutz Strüngmann (2002)
Rendiconti del Seminario Matematico della Università di Padova
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Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2015)
Formalized Mathematics
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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].