Exercise 6, § 3

(a) Let A be a valuation ring. Extend to finitely generated A-modules the results of Algebra, Chapter VII, § 4, nos. 1 to 4 (theory of invariant factors).

(b) Let K be a field, $v$ a valuation on K, A the ring of $v$ and $\Gamma$ the order group of $v$. Let $U = (\alpha_{ij})$ be a square matrix of order $n$ with elements in A and of determinant 0; show that there exists $\lambda \in \Gamma$ such that, for every square matrix $U' = (\alpha'_{ij})$ of order $n$ with elements in A and satisfying the conditions $v(\alpha'_{ij} - \alpha_{ij}) > A$ for every ordered pair $(i, j)$, the invariant factors of $U'$ are the same as those of $U$.

(c) Under the hypotheses of (b), generalize the following results of Algebra, Chapter IX, § 5, no. 1, Theorem and Exercise 1 and § 6, Exercise 10 (for the last, suppose that K is not of characteristic 2 and that $v(2) = 0$).