Lemma 3

Let $A$ be a complete Hausdorf topological ring, in which there exists a fundamental system $\mathscr{G}$ of neighbourhoods of 0 consisting of additive subgroups of $A$. (i) For all $x \in A$ such that $\lim_{n \to m} x^n = 0$, $1 - x$ is invertible in $A$ and its inverse is equal to $\sum_{n=0}^\infty x^n$. (ii) Let $a$ be a two-sided ideal of $A$ such that $\lim_{n \to \infty} x^n = 0$ for all $x \in a$. For an element y of $\mathbf{A}$ to be invertible, it is necessary and sufficient that its class mod. a be invertible in $\mathbf{A}/a$; in particular a is contained in the Jacobson radical of $\mathbf{A}$.

(i) As

it all amounts to proving that the series with general term $x^n$ is convergent in $\mathbf{A}$; now, by hypothesis, for every neighbourhood $V \in \mathfrak{S}$ of 0 in $\mathbf{A}$, there exists an integer $p > 0$ such that $x^n \in V$ for all $n \geq p$. We conclude that

for all $q \geq p$ and our assertion then follows from Cauchy's criterion (General Topology, Chapter III, § 5, no. 2, Theorem 1).

(ii) Suppose that there exists $y' \in \mathbf{A}$ such that $yy' \equiv 1$ (mod. a) and $y'y \equiv 1$ (mod. a). The hypothesis on a implies, by (i), that $yy'$ and $y'y$ are invertible in $\mathbf{A}$ and hence $y$ is invertible in $\mathbf{A}$. In particular, every $x \in a$ is such that $1 - x$ is invertible in $\mathbf{A}$ and, as a is a two-sided ideal of $\mathbf{A}$, it is contained in the Jacobson radical of $\mathbf{A}$ (Algebra, Chapter VIII, § 6, no. 3, Theorem 1).

Having established this lemma, it is sufficient to apply it to the topological ring $\hat{\mathbf{A}}$ and the ideal $\hat{\mathfrak{m}}$, as, for all $x \in \hat{\mathfrak{m}}$, $x^n \in (\hat{\mathfrak{m}})^n \subset (\mathfrak{m}^n)^{\wedge}$ and the sequence $(x^n)$ therefore tends to 0.

If we take $\mathbf{A} = \mathbf{Z}$, every maximal ideal of $\mathbf{Z}$ is of the form $p\mathbf{Z}$ where $p$ is prime. The ring of $p$-adic numbers $\mathbf{Z}_p$ is then a local ring of which $p\mathbf{Z}_p$ is the maximal ideal (Corollary 2 to Proposition 16) and whose residue field is isomorphic to $\mathbf{Z}/p\mathbf{Z} = \mathbf{F}_p$, and $\mathbf{Z}_{(p)}$ with the $p\mathbf{Z}_{(p)}$-adic topology is identified with a topological subring of $\mathbf{Z}$, containing $\mathbf{Z}$.