Corollary 3

(i) Let $b$ be an ideal of $A$ and $a$ an element of $A$. For $a \in b$, it is necessary and sufficient that, for all $m \in \Omega$, the canonical image of $a$ in $A_m$ belong to $bA_m$.

(ii) In particular, let b and c be two elements of A. For c to be a multiple of b, it is necessary and sufficient that, for all $m \in \Omega$, the canonical image of c in A, be a multiple of that of b.

As $bA_m = b_m$ (§ 2, no. 7, Corollary to Proposition 18), (i) is a special case of Corollary 1; (ii) follows from (i) applied to the ideal $Ab$.