A Note on Finitely Generated Z-module and Algebraic Integers

The theory of algebraic integer has its many applications, such as in algebraic coding, cryptology, information system and other fields. The research of algebraic integer can not leave finitely generated module, and the finitely generated module itself be also applied in group theory, ring theory, and some applied science. In this paper, we research the theory of algebraic integer using finitely generated module as tool, we obtained necessary and sufficient condition that an element is algebraic integer, and an intrinsic connects between algebraic number field and finitely generated Z-module.