Exercise 12, § 3

(a) Let $A$ be a commutative Noetherian ring and $m_1, m_2$ two ideals of $A$ contained in the Jacobson radical of $A$; let $A_1$ be the completion of $A$ with respect to the $m_1$-adic topology. If $m_1 \subset m_2$, show that the identity mapping on $A$ can be extended by continuity to an injective representation of $A$, into $A_2$ (cf. General Topology, Chapter III, § 3, no. 5, Proposition 9).

(b) Let $n = A_1 m_2$. Show that, if $A_1$ is canonically identified with a subring of $A_1$, $A_1$ is identified with the completion of $A_1$ with respect to the $n$-adic topology.