Exercise 18, § 4

Let A be a Dedekind domain, K its field of fractions and M, N two lattices in a vector space V of finite rank $n$ over K. Let $e_k = e_k(N, M)$ be the invariant factors of N with respect to M (Exercise 14). Show that there exists a basis $(u_i)_{1 \leq i \leq n}$ of V such that M is equal to a direct sum $\bigoplus_i a_i u_i$, where the $a_i$ are fractional ideals, and N is equal to the direct sum $\bigoplus_i e_i a_i u_i$. (Use the theory of finitely generated modules over a principal ideal domain (Algebra, Chapter VII, § 4, no. 2, Theorem 1) and the approximation theorem in the unimodular group $\mathbf{SL}(n, A)$ ($§ 2$, no. 4).)

Extend this result to the case where A is the union of a right directed family of subrings which are Dedekind domains (for example the integral closure of a Dedekind domain in the algebraic closure of its field of fractions).