Exercise 26, § 2

Let $A$ be a commutative ring, $m$ an ideal of $A$, $(\mathfrak{a}_\lambda)_{\lambda \in L}$ a system of generators of $m$ and $\hat{A}$ the Hausdorff completion of $A$ with respect to the $m$-adic topology.

Show that, if $A'$ denotes the ring of formal power series in a family $(T_\lambda)_{\lambda \in L}$ of indeterminates with only a finite number of terms of 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, $- a_\lambda$ (use Theorem 1 of no. 8). In particular, the ring $\mathbf{Z}_p$ of $p$-adic integers is isomorphic to the quotient $\mathbf{Z}[[T]]/(T-p)$.