习题 2, § 2

设A为主理想整环，设π为A的不可约元，设M为π主模；证明对于所有 x ∈ M 和所有 a ∈ A 而不是 π 的倍数，存在唯一的 y ∈ M 使得 x = αy（使用 Bezout 恒等式）；设 y = $\alpha^{-1}x$。 Deduce that, if $A_{(\pi)}$ denotes the principal ideal domain consisting of elements $\beta/\alpha$ of the field of fractions of A, such that α, β ∈ A and α is not a multiple of π (VII, p. 48, Ex. 4), then there exists a unique $A_{(\pi)}$-module structure on M such that restriction of the ring of operators to A gives the original A-module structure on M. 由此定义的 $A_{(\pi)}$ 模块据说与 M 规范相关；它的子模与M的A-子模相同。