Corollary 1

Let $A$ be a commutative Noetherian ring, $m$ an ideal of $A$, $E$ a finitely generated $A$-module and $F$ and $G$ two submodules of $E$. Let $A, E, F, G$ be given the $m$-adic topologies and let $i$ be the canonical mapping from $E$ to $e$. Then:

Moreover, if $a$ and $b$ are two ideals of $A$ and $c = ab, \hat{c} = \hat{a}\hat{b}$.

By Theorem 3, $\hat{E}, \hat{F}, \hat{G}$ are canonically identified with $\hat{A} \otimes_A E, \hat{A} \otimes_A F, \hat{A} \otimes_A G$, which establishes the first two formulae. The third and fourth follow respectively from Chapter I, § 2, no. 6, Proposition 6 and no. 10, Proposition 12. Finally, as $\hat{a} = \hat{A}i(a), \hat{b} = \hat{A}i(b), \hat{c} = \hat{A}i(c)$,