Exercise 26, § 2

Let $A$ be a commutative ring, $m$ an ideal of $A$, $(a_\lambda)_{\lambda \in L}$ a system of generators of $m$, $\hat{A}$ the separated completion of $A$ for the $m$-adic topology.

Now prove that, if $A'$ denotes the ring of formal series with respect to a family $(T_\lambda)_{\lambda \in L}$ of indeterminates, having only a finite number of terms of a given degree (Exercise 12), $\hat{A}$ is isomorphic to the quotient of $A'$ by the closure $b$ of the ideal of $A'$ generated by the $T_\lambda - a_\lambda$ (use Theorem 1 of No. 8). In particular, the ring $\mathbf{Z}_p$ of $p$-adic integers is isomorphic to the quotient $\mathbf{Z}[[T]]/(T-p)$.