@MISC{ZMODUL05.MIZ,
  AUTHOR = {Nakasho, Kazuhisa and Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
  TITLE = {Rank of Submodule, Linear Transformations and Linearly
    Independent Subsets of $\mathbb Z$-module},
  SECTION1 = {Rank of Submodule of $\mathbb Z$-module},
  SECTION2 = {Basic Facts of Linear Transformations},
  SECTION3 = {Some Basic Facts about Linearly Independent
    Subsets and Linear Combinations},
  DAY = {10},
  MONTH = {July},
  YEAR = {2014},
  ADDRESS1 = {Shinshu University, Nagano, Japan},
  ADDRESS2 = {Japan Advanced Institute of Science and Technology, Ishikawa, Japan},
  ADDRESS3 = {Shinshu University, Nagano, Japan},
  ADDRESS4 = {Shinshu University, Nagano, Japan},
  SUMMARY = {In this article, we formalize some basic facts of $\mathbb Z$-module.
  In the first section, we discuss the rank of submodule of $\mathbb Z$-module and its properties.
  Especially, we formally prove that the rank of any $\mathbb Z$-module
  is equal to or more than that of its submodules,
  and vice versa, there exists a submodule with any given rank that satisfies the above condition.
  In the next section, we mention basic facts of linear transformations between two $\mathbb Z$-modules.
  In this section, we define homomorphism between two $\mathbb Z$-modules
  and deal with kernel and image of homomorphism.
  In the last section, we formally prove some basic
  facts about linearly independent subsets and linear combinations.},
  ACKNOWLEDGEMENT = {This work was supported by JSPS KAKENHI
  21240001 and 22300285.}}

@ARTICLE(LAT,
      AUTHOR  = {Daniele Micciancio and Shafi Goldwasser},
      TITLE   = {Complexity of Lattice Problems:
      A Cryptographic Perspective (The International Series
      in Engineering and Computer Science)},
      PUBLISHER={Springer},
      YEAR    = 2002)
