Exercise 2, § 2

Let A be a principal ideal domain, let π be an irreducible element of A and let M be a π-primary module; show that for all x ∈ M and all a ∈ A not a multiple of π, there exists a unique y ∈ M such that x = αy (use the Bezout identity); put 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. The $A_{(\pi)}$-module thus defined is said to be canonically associated to M; its submodules are identical to the A-submodules of M.