Example 5

Let $\mathfrak{a}$ be a two-sided ideal of A. The $\mathfrak{a}$-adic topology on A is the topology, compatible with the ring structure of A, for which the ideals $\mathfrak{a}^n$ (for $n\geqslant 1$) form a fundamental system of neighborhoods of 0 (Gen. Top., III, §6, No. 3, p. 275, Example 3). Suppose that the ring A is Hausdorff and complete (Gen. Top., III, §6, No. 5, p. 276) for this topology; this is, for example, the case when the ideal $\mathfrak{a}$ is nilpotent. For every $x\in \mathfrak{a}$, the series $\sum^{\infty}_{n=0}(-x)^n$ is then convergent. Let $y$ be its sum. We have $y-1 =\sum^{\infty}_{n=1}(-x)^n=-xy$, hence $(1 +x)y= 1$. Likewise, we have $y(1 +x) = 1$, so $1 +x$ is invertible. By Theorem 1, it follows that the ideal $\mathfrak{a}$ is contained in the radical of A.